O. Kusch

Computer-Aided Optical Design

of Illuminating and Irradiating Devices

«ASLAN» Publishing House Moscow 1993

2

O.K. Kusch Computer-Aided Optical Design of Illuminating and Irradiating Devices

ISBN 5-87793-001-X

© O.K. Kusch, 1993 translated by V.N. Stepanov

Христианско-просветительское издательство «Аслан» Москва 125167, Ленинградский проспект, 60-67 тел. (095) 151-52-28

_________________________________________________ Подписано к печати 05.12.1993. Формат бумаги 60х88 1/16. Гарнитура «Таймс». Печать офсетная. Авт.л. 11,00. Физ.печ.л.12,00. Тираж 1500 экз. Христианско-просветительское издательство «Аслан». Изд. Лицензия ЛР № 062738

Отпечатано в Подольском филиале Чеховского полиграфического

комбината 142110, г.Подольск, ул.Кирова, 25

3CONTENS INTRODUCTION 5 CHAPTER ONE GEOMETRICAL OPTICS OF REFLECTORS AND REFRACTORS 1.1. Elements of Linear and Matrix Algebra 9 1.2. Surface Theory Elements 15 1.3. Laws of Reflection and Refraction. (Snell’s Laws) 21 1.4. Ray and Surface Intersection 27 1.5. Ray Tracing for Discrete-Point Representation of a Surface 32 1.6. Surface Synthesis by Using Fermat’s Principle 35 1.7. Matrix Application in Geometric and Optical Calculations 42 References 50

CHAPTER TWO

METHODS FOR CALCULATING LIGHT DISTRIBUTION

2.1. Light-Ray Methods 51 2.2. Light Distribution Created by Ellipsoid or Hyperboloid With a Point Source 58 2.3. Inverse-Ray Methods 64 2.4. Inverse-Ray Method in Analytical Representation 83 2.5. Calculation of Illuminance by Wiener’s Scheme 99 2.6. The Method of Elementary Maps 107 2.7. Probabilistic Simulation in Design of Lighting and Optical Fixtures 112

4

2.8. Calculation of Light Distribution from Lambertian Sources 119 2.9. Models of Real Sources and Materials 129 2.10. Graphical Representation of Luminous Field Produced by Lighting Fixture 135 2.11. Calculating Light Source Indicatrix for Specified Luminous Intensity 138 References 142 CHAPTER THREE INVERSE PROBLEM IN OPTICAL SYSTEM DESIGN 3.1. Point-Source Methods and Algorithms 145 3.2. Methods and Algorithms of Solving Inverse Problem for Lengthy Sources 154 3.3. Calculating Schemes for Reflector Optimization 166 3.4. Optimization of Focal Parameter of Reflector 168 3.5. Calculation of Milling-Cutter Trajectory 170 References 172 ANNEXES Annex 1 173 Annex 2 177 Annex 3 182 Annex 4 188

5

To the memory of V.D. Komissarov and N.G. Boldyrev — remarkable Russian mathematicians and illuminating engineers.

INTRODUCTION

The computers, presently, have germinated new opportunities in design of illuminating and irradiating devices. These «clever» machines led to improvement of «old» and stimulated the birth of new methods in design of optical systems for illumination and irradiation. Renewal of old undeservingly forgotten methods is not a tribute to dull tradition, but it is governed by intention to find the most efficient practice adequate to rapid development and broad application of lighting devices with various light sources. It should be noted that Russian school has deep roots in this field. V.N.Chikolev established first bricks at the theory of design of lighting fixtures on the eve of the century. Fundamental works were published in 20-40s by V.A.Fock, N.G.Boldyrev, A.A.Gershun, V.D.Komissarov, and in 50-70s by N.N.Ermolinskii, N.A.Karyakin, V.V.Trcmbach. It is of great importance to follow the ideas that could have made a link between different methods that exist today. To author’s mind, the principles of geometrical optics and light field theory* can serve this uniting base.

Of course, geometrical and optical principles of design and calculation should be set forth in the modern language of vector-and-matrix algebra.

Fermat’s variational principle is one of the basic statements in geometrical optics. It enables to synthesize optical systems most easily and elegantly.

«The point source» is also one of the main concepts in geometrical optics. Its role is similar to that of «the point mass» in theoretical mechanics. Distribution of point sources in space is interpreted in modern theory as a distribution of Dirac’s δ-functions. The point source concept enables to connect differential properties of an optical surface, Gaussian curvature, in particular, with distribution of reflected intensity.

Another important concept in optical calculations is the ray since it terms all photometric values that affect a detector being sensitive to quadratic constituent of a light field: illuminance, spherical illuminance, luminous

* The author is familiar with similar views of Dr. O.N.Stavroudis and Dr. S.Cornbleet on fundamental role of geometrical optics in optical systems design.

6

intensity, etc. Fock and Wiener’s theorems arc fundamental in calculation of intensity.

It seemed very important to classify the problems of optical design by separating them in two groups: direct and inverse problems.

Determination of intensity* in luminous field produced by a luminaire at the proximate zone (the illuminance) or at the remote zone (the luminous intensity), when geometrical and luminance parameters of the optical system arc known, constitute the direct problems (the analysis problems).

Following Baltes**, we can describe an optical system by a set of space-and-luminance characteristics },...,,{ 21 nLLLL , while a resulted intensity distribution by a set },...,,{ 21 nEEEE , we specify the direct problem in terms of mapping F: L→E. The inverse mapping F-1:E→L, generally, is a solution of inverse problem. In the theory of light-fixtures design an inverse problem means the determination of mirror surface shape that provides a prescribed intensity distribution and meets a chosen optimizing criteria. Inverse problems usually lead to the necessity of solving integral and differential equations.

There are two approaches to solving direct problems. The first deals with computation of intensity within a ray tube related to a point source. The result is considered as a superposition of intensities produced by these point sources. The second approach is based upon calculation of ray luminance while tracing the rays through the optical system and applying the laws of geometrical optics and the most general equations of light-field theory. By tracing a multitude of rays, the luminance at the points of exit pupil of a fixture, and consequently, in accordance with Mangen’s law, a desirable intensity characteristic can be determined.

Elements of linear and matrix algebra as well as foundations of surface theory are given in the First Chapter. The laws of geometrical optics, the laws of reflection and refraction, Fermat’s principle and its applications in synthesis of optical surfaces arc to be found here.

The Second Chapter deals with the methods of calculating luminous distribution of illuminating and irradiating dcviccs, i. c. with direct problems and their solutions. The main equations of ray method based on the point-source conception are cited. The ideas of inverse-ray method are described in details. The latter being the most accurate method for calculating various optical systems with sources of finite dimensions. Its analytical version enables * Under the term «intensity» we mean power characteristics of a field: illuminance, irradiance, or luminous

intensity. ** Inverse Source Problems in Optics, lid. by 11.P. Unites with a Foreword by J.-F. Moser. 1984.

7

to construct effective algorithm and better understand concentrating properties of specular reflectors.

The method of elementary maps, which is being used in Russian school of illuminating engineering when dealing with Lambertian sources, has been also incorporated. Examples on application of contour integration and light-field methods developed by Boldyrev and Gershun are presented as well.

The most notable addition herein is the paragraph on direct statistical simulation of lighting fixtures, i.e. on application of Monte Carlo methods.

The problem has been stated on restoration of local characteristics of radiation in a light source through its integral output (the inverse problem). This is actual, since it enables to define the input parameters more exactly, and thus to increase the accuracy of direct problem solution.

The subject matter in the Third Chapter touches theoretical and practical spectrum of inverse problem. Classical results for point and linear sources are presented. The conditions of algorithm convergence within a domain being free from critical points have been analyzed. The solutions for the sources with finite dimensions arc given.

Annexes contain the texts of C-programs for solving various direct and inverse problems (solution of nonlinear equation, spline interpolation, integration of a system of differential equations).

The author hopes that this book will serve to invoke new ideas and unite the optical design of lighting devices with the general methodology of modern optics and illuminating engineering.

The author expresses deep gratitude to V.N. Stepanov who translated the book and assisted in its publishing.

Author

8

9

CHAPTER ONE GEOMETRICAL OPTICS OF REFLECTORS AND REFRACTORS 1.1. ELEMENTS OF LINEAR AND MATRIX ALGEBRA

For unification wc shall use the matrix notation accepted in courses for

matrix algebra [1]. A vector is represented as a column [2]

nx

xx

x

.

.

.2

1

The operation of transposition, which changes columns for rows and vice versa, is denoted by upper index T. In that representation quantities xi are the coordinates of vector x, and the latter will be written in a matrix form:

Tnxxxx ...21 Vector is usually expressed via basis: we choose a conventional set of

vectors e1, e2, e3,..., en and express all other vectors as

n

iiinn exexexexx

12211 ... (1.1)

Basis is a system of linearly independent vectors, hence, 01

n

iiiea that

leads to ai=0 for ni ,1 . Basis vectors form a coordinate system, where numbers are projections of vector on corresponding axes.

10

The scalar product of vectors x and y is

n

iii

T yxyxyxyx1

),( ,

where xi and yi, are the coordinates of vectors x and y. The length (absolute value) of a vector is computed as

n

ii

T xxxx1

22.

Usually, basis is formed by a set of mutually orthogonal unit vectors niei ,1,ˆ , for which the following equalities are valid

ijjT

i ee ˆˆ ;

n

i

ijj ee

1

221)(ˆ ,

where ij , is Kronecker’s symbol, ij = 1 for i=j and ij = 0 for i≠j. For example, in three-dimensional space the system of vectors

Te ]001[1 , Te ]010[ˆ2 , Te ]100[ˆ3 forms the basis. Really,

0100

010

001

3

2

1

221

,

hence α1=α2=α3=0 Matrix A describes linear mapping of vector x into vector y that can be

written as y = Ax. The linearity of transformation A means that the following equalities hold true

AxxA )( ,

2121 )( AxAxxxA . Substituting decomposition of x from Eq. (I.I), we get

i

ii eAxAxy ˆ .

Now, ieA ˆ , is a new vector, which can be decomposed in the form

i

ijii eeA ˆˆ (decomposition by «columns» of matrix A).

11

By changing the order of adding in the sums, we get

iijii

i iiji

iiji

ii eyexaeaxy ˆˆˆ ,

hence,

i

ijii xay . (1.2)

Thus, (he coordinates of vector y, unlike for the transformation of basis vectors, are obtained by multiplying rows of matrix A by the column-vector [x1 x2…xn]T.

Matrix A is defined as

mnmm

n

n

aaa

aaaaaa

A

21

22221

11211

.

.

.......

.

Matrix A = {aij} has m rows and n columns. Further, we introduce a unit matrix I, for which AI=IA=A, and an inverse

matrix, such that A-1 A=I. The rules for operating with matrices follow from the properties of linear

operators. For example, the rule of multiplication of matrices can be obtained if the rule for multiplication of operators is applied:

)()( BxAxAB . The k-th element of vector Bx is

jjkj xb , [see Eq. (1.2)], then the i-th

element of vector A(Bx) will be

k

kjikj

jj

jkjk

ik baxxba .

Thus, it follows that the element of a new matrix C is a product of the i-th row of matrix A by the j-th column of matrix B:

k

kjikij bac (1.3)

Let us now consider how vector coordinates and linear operator components arc transformed if basis is changed. We have a basis { '

ie } expressed in terms of unprimed system by relation

12

j

jjii ee ' ,

where n2 of elements ji evolve the coefficients of transforming matrix that carries out the transition from one coordinate system to another.

Let us consider an arbitrary vector having coordinates xi and 'ix in two

systems, respectively. Then

i

ijij

jj

jjii

iii

i xeexexx '''' ;

hence,

i

ijij xx ' ,

and that is equivalent to matrix product `xx . (1.4)

Multiplying Eq. (1.4) by inverse matrix γ-1, we obtain xx 1` . (1.5)

Orthogonal matrix P can be defined from the condition IPPPP TT ,

where I is the unit matrix. It is an obvious property of such matrices that P-

1= PT, hence, Eq. (1.5) takes the form xx T . (1.6)

Operators also change their form during transition to another basis. Let y=A`x` and y=Ax in two different systems, then

`` 111 xAAxyy . Consequently, the rule for matrix transformation will take the form

AA 1` . (1.7) Note, that during the transition to another basis, the determinant does not

change, i. e. det(A')=det(A). We also emphasize the rule: if C = AB, then

TTTT ABABC )( . (1.8) Expression (1.8) directly happens out from Eq. (1.3). Let us now consider

few examples.

13

Example I. Matrix of Rotation We choose on a plane a system

of two orthogonal vectors Te ]01[1 , Te ]10[ˆ2 and rotate

it by an angle θ anticlockwisely (Fig. 1.1). In accordance with the rule of constructing matrix of linear transformation we find the transformed vectors of the basis:

cossin

ˆˆ

sincos

ˆˆ

22

11

eAe

eAe,

Expressions being found are the columns of matrix of rotation Mrot: Mrot:

cossinsincos

rotM . (1.9)

Matrix of rotation Eq. (1.9) is orthogonal, because after rotation by angle -θ (clockwisely), vector x returns to its initial position. Sign change from + to - in Eq. (1.9) shifts orthogonal matrix into transposed one. Matrices of rotation on the plane commutate, i. e.

)()()()( 1221 rotrotrotrot MMMM . Rotation is unitary, i. e.

1)det( rotM . For example, let us rotate a triangle ABC by 90° about the origin of coordinates (Fig. 1.2) [3], i. e.

12

14

13

0110

21

41

31 .

Fig. 1.1. Rotation of unit square

Fig.1.2. Two-dimensional rotation by 90°

14

Example 2. Matrix of Reflection We consider mirror reflection with

respect to bisectrix y = x. It is easy to show for this case that Te ]10[`ˆ 1 ,

Te ]01[`ˆ 2 , and the law of reflection is defined by the matrix

0110

A .

Let us find mirror image of triangle ABC (Fig. 1.3):

62

73

81

26

37

18

0110

Similar to matrix of rotation, matrix of reflection does not change metrical properties of figures, i. e. 1)det( A .

Generally speaking, when applying operator A to vector x, we get vector Ax, which differs from x. Vector, for which Au = λu is called an eigenvector u with eigenvalue λ. In case of space rotation about axis OZ, any vector directed along the axis of rotation will be an eigenvector with eigenvalue of +1.

Let us assume eigenvectors ie , as a basis for matrix A, that is,

j

jjii eaeA ˆˆ

and

iiii

iii exeAxy ˆˆ ,

then j

jjji eea ˆˆ .

Thus,

jijii

a ji ,0, .

Matrix A takes, so called diagonal form: numbers λi, stand along its diagonal, while the rest places are occupied by naughts.

From the previous example we define the matrix of reflection if we consider a new basis, which is being rotated in relation to the initial basis by the angle of 45°:

Fig. 1.3. Two-dimensional reflection with respect to y=x

15

22

22

22

22

.

Wc carry out multiplication of matrices in accordance with the law of similarity [see Eq. (1. 7) ]:

1001

22

22

22

22

1001

22

22

22

22

1 A .

Obviously, there arc two eigenvectors of reflection: the vector directed along the straight line y = x, and the vector, which is orthogonal to this line.

1.2. ELEMENTS OF SURFACE THEORY

Parametrical representation is the most advisable way for description and

study of surfaces. Let there be a vector-function of two scalar variables within Cartesian basis kij ˆ,ˆ,ˆ :

kvuzjvuyivuxvurr ˆ),(ˆ),(ˆ),(),( . A tangent vector to the curve r =

r(u,vo), where v0 is a constant, is a multiple to vector urru / . Similarly, a tangent vector to the curve r=r (u0, v) is a multiple to vector vrrv / . A plane, which is tangent to these curves (parametric curves) contains both said vectors, therefore, a normal to the surface at the point uo,vo is a multiple to their vector product (Fig. 1.4). A unit normal vector is expressed as

vuvu rrrrn */]*[ˆ . (1.10) Matrices of the first G and the second

D fundamental forms

Fig.1.4. Tangent vectors and coordinate lines ru and rv

16

[4] play a substantial role in differential geometry of surfaces:

,ˆˆˆˆ

,2

2

vvuv

vuuu

vuv

vuu

rnrnrnrn

G

rrrrrr

G, (1.11)

where 22 / urruu ; vurruv /2 .

An area of surface clement is equal to the area of parallelogram constructed on vcctors ru∆u and rv∆v , i. e.

vurrS vu * . We note, that

Gggggrrrrrr vuvuvu 21122211222 )(* ,

where gij are the elements of the first fundamental form and )det(GG . An area of surface specified by the domain R of variables u, v can be calculated as follows

R

dvduGS 2/1 .

A curve on the surface r (u,v) can be specified when parameters u and v are replaced by a pair of functions in terms of a new parameter, say t: u=u(t), v = v(t) or by expression U = U(t), where TtvtuU )]()([ .

A tangent vector to this curve is UAvrurr vu , (1.12)

where

vz

uz

vy

uy

vx

ux

A .

When the surface is intersected by a plane containing the main normal

on and the tangent vector UAr Fig. 1.5), a curve being

17

produced is called the normal section. A normal curvature kn of a surface is defined as follows [4]

UGUUDUk T

T

n

. (1.13)

For any other curves having at a given point the similar direction and the normal no the theorem of Meusner is valid (Fig. 1.5)

.cos constkk n The directions, for which the normal

curvature is minimum or maximum, are termed the principal directions. From Eq. (1.13), by differentiating with respect to u and v , we obtain

,0)()(,0)()(

22222121

12121111

vgkdugkdvgkdugkd

nn

nn

(1.14) By eliminating u and v from the equations, we find

0)2( 1212221122112 DkdggddgkG nn , (1.15)

whence, the minimum and maximum curvature values can be calculated. The principal directions in curvature are available when kn is eliminated from Eq. (1.14). We arrive to quadratic equation

vugdgdugdgd )()( 112222112

11121211 0)( 2

12222212 vgdgd , (1.16) whence, the ratio vu : for the principal directions is determined. From Eqs. (1.12) and (1.13) it follows that k1≠k2 then the principal directions arc orthogonal.

Calculation of Gaussian and mean curvature is of special interest:

)(21, 2121 kkHkkK .

In agreement with the features of the roots of Eq. (1.15)

Fig. 1.5. Tangent vector UA to curve C at point M; normal section Co; curvature

radius R=Ro cos θ

18

121222112211 22, dggddgHGD

K . (1.17)

Example Let a surface be formed by rotation of a curve C

)(),( uzux lying in the plane XOZ, about the axis OZ (Fig. 1.6). Let v be an angle of rotation. The expression for the surface of revolution is

ivur ˆcos)(

kujvu ˆ)(ˆsin)( or

kuveur ˆ)()(ˆ)( . (1.18)

where jvivve ˆsinˆcos)(ˆ is the unit vector. The lines of v=const are the meridians in the surface, and the lines of u =const are the parallels. Now the elements of matrices G and D can be found. Differentiating Eq. (1.18), we have

).(ˆ)();2/(ˆ)(

;ˆ)``()(ˆ)``(

;ˆ)`()(ˆ)`(

veurveur

kuveur

kuveur

v

v

uu

u

Whence,

.)``()(

);(

;0;``

2/1222/12211

2/1

2222

12

22211

ggG

urgrrg

rg

v

vu

u

Thus, the meridians and parallels form orthogonal grid on the surface of revolution (g12 = 0).

We find the normal vector and coefficients dij

Fig. 1.6. Curvalurc lines on a surface of revolution

19

2/1222/1 )``()(ˆ`ˆ`

ˆˆˆ

ˆ

vek

G

zyxzyxkji

nvvv

uuu

;

.)```(

;0ˆ

;)``(

``````ˆ

2/12222

12

2/12211

dnrd

nrd

uv

uu

(1.19)

Determining the principal directions from Eq. (1.14), we obtain

022112211

vuggdd

hence, ou or ov , i. e. the principal directions coincide with meridians and parallels on the surface of revolution. Assuming in Eq. (1.14) first ov (the meridian direction), and then ou (the parallel direction), we have, correspondingly

.)``(

`

,)``(

``````

2/12222

222

2/32211

111

gdk

gdk

(1.20)

We choose the curve x=x(z) (u=z) for meridian. From Eqs. (1.20) we have

2/321 )`1(``

xxk

, 2/122 )`1(

1xx

k

.

From geometrical considerations it is obvious that the curvature of normal section is equal to the meridian curvature. Since

dxxxkd ,cos/)`1(/1 2/122 is the length of normal segment

between the surface and the intersection point of normal with the axis (Fig. 1.6).

Let us calculate Gaussian and mean curvatures:

.)``(

)```(``)````(5.0

,)``(

``)`````(

2/322

22

222

H

K (1.21)

20

An important specific case should be noted, when a surface is formed by rotation of a curve termed in polar coordinates R=R(u). Here φ(u)=R sin u, ψ(u)=R cos u. Further, we find

2/1222/1 )(sin uRRuRG . By definition of derivative

RRu u /)(tan , where δ is an angle between the normal and the axis.

Thus, )cos(/sin22/1 uuRG . (1.22)

and the normal vector Tvvn ]cossinsincos[sinˆ . (1.23)

Using Eq. (1.20), the curvatures in meridian and sagittal sections can be found

.sin

)(sin,)(

222/322

22

1 uruk

RRRPRRk

u

uuu

(1.24)

The latter equation has an obvious geometrical meaning and can be derived directly from Meusner’s theorem.

The divergence of a light beam which is reflected from a mirror segment, as it will be shown below (see Ch. 2), depends on the principal curvatures at the point of reflection. We find these values for paraboloid of revolution. A parabola equation in polar coordinates can be termed as follows

)2/(cos)cos1/(2 2 ufufr , (1.25) where f is the focal distance of parabola, and u is counted from the focal axis. The principal meridian curvature can be conveniently calculated when applying Eq. (1.24). Differentiating Eq. (1.25) and substituting the result into Eq. (1.24), after transformations, we obtain

).2/(cos)2/1( 31 ufk (1.26)

The curvature in the other principal section can be found easily by using Meusner’s formula

).2/cos()2/1(2 ufk (1.27)

21

1.3. LAWS OF REFLECTION AND REFRACTION (SNELL’S LAWS)

The Law of Reflection Let oa be the direction of incident ray, a be the direction of reflected ray,

and n be the normal to the surface at the point of incidence (Fig. 1.7 a). Since

oa , a , and n are the unit vectors, the law of reflection is represented as follows:

(1) the vectors oa , a , n are coplanar, i. e. they lie in one plane; (2) the angle of incidence is equal to the angle of reflection, i. e. the

following equation is true .ˆˆˆˆ nanao (1.28)

In linear algebra terms the first statement means that oa , a , and n are linear-dependent. Therefore, the scalar quantities λ, μ, ν exist such that they are not equal to zero simultaneously; this leads to

.0ˆˆˆ nvaao (1.29) Since Eq. (1.28) is symmetrical, Eq. (1.29) can be represented in the form

naao ˆˆˆ , (1.30) where γ is a new scalar constant.

Fig. 1.7. On deduction of laws of reflection (a) and refraction (b) in vector form

22

Multiplying Eq.(1.30) by n scalarly, and taking account of Eq. (1.28), we get

nao ˆˆ2 . Consequently, the law of reflection is defined as follows

)ˆ,ˆ(ˆ2ˆˆ nanaa oo . (1.31) Equation (1.31) allows to solve the initial problem — to find the direction

of the reflected ray when the incident ray and the normal at the mirror’s point are known. Here Eq. (1.31) is inversible. If we change a for oa equation (1.31) remains unchanged. This allows to analyze a mirror fixture either in direct or inverse ray-tracing mode (sec Ch. 2).

The vector operation in Eq. (1.31) can be represented in the matrix form oaEIa ˆ)2(ˆ , (1.32)

where

;100010001

I .

2

2

2

zyzxz

zyyxy

zxyxx

nnnnnnnnnnnnnnn

E

We put the law of reflection as follows

Tozoyox

zyzxz

zyyxy

zxyxxT

zyx aaannnnn

nnnnnnnnnn

aaa ][*1222

21222212

][2

2

2

,

(1.33) or, denoting the matrix as Mrefl,

oreflaMa ˆˆ . The matrix Mrefl depends on basis choice. In the natural basis connected

with the point of ray incidence, where the axis OZ coincides with n , the matrix Mrefl, as it can be easily checked, will take a simple diagonal form.

Really, it follows from Eq. (1.31) that oa = n is an eigenvector of the linear transform Mrefl with the eigenvalue of -1, and any vector, which is orthogonal to n , is an eigenvector with the eigenvalue of +1. Hence, the matrix Mrefl takes the form

23

100010001

reflM . (1.35)

The main advantage of using matrices is the possibility of multiplying them, so that computing automation for the ray path through several reflecting surfaces can be realized [5]. For instance, for the double specular reflection the vector coordinates are determined by matrix product

)2)(2( 21 EIEIM refl , (1.36) where E1 and E2 are matrices related to the normal vectors of the first and the second mirrors, correspondingly.

The Law of Refraction According to the first law of rcfraction the vectors oa , a , and n are

coplanar (Fig. 1.7b), i.e. it is true that ,ˆˆˆ naa o (1.37)

where λ, μ are the constants to be determined. The second law of refraction stales that q sin α=sin β, where q=v1/v2 is the

relative refractive index, and can be termed in vector form as follows ]ˆˆ[/1]ˆˆ[ 0 naqna . (1.38)

Multiplying Eq. (1.37) by n vectorially, and taking account of Eq. (1.38), we find that λ=q. We rewrite Eq. (1.37) as follows

,ˆˆˆ naqa o (1.39) Since oa , a , and n are unit vectors, the square of the above equation

yields ,0)1(2 22 qqp

where p= oa * n . Its solution is

)1()( 22 qqpqp . The law of refraction is represented in the form

,ˆ}])ˆ,ˆ(1[1ˆˆ{ˆˆ 22 nnaqnaqaqa ooo (1.40)

24

Fig. 1.8. Ray tracing through system of refracting surfaces:

a - incident and reflected rays; b - propagation of ray after refraction

The formulas in Eqs. (1.31) and (1.40) arc applicable for programming a computer. Only the following has to be noted. The equation (1.40) docs not

hold true when 2)ˆ,ˆ(1/1 naq o *, this relates to the total internal reflection. The upper sign before square root has been choscn in Eq. (1.40); it can be proved when noting that for q= -1 Eq. (1.40) has to describe the law of reflection [see Eq. (1.31) ].

Evidently, Eq. (1.40) is nonlinear, so it can not be presented in the matrix form. But if we take bounded range of small angles α, so that sin α ≈ α, the law of refraction transforms into linear form, and use of matrices becomes possible.

A paraxial ray in the system with rotational symmetry is defined by two parameters: the linear coordinate, that is the height where the ray intersects the reference tangent plane; and the angle α between the ray and the axis (Fig. 1.8 a). Let the ray coordinates hk-1 and α k-1 corresponding to the (k-l)-th plane be known. Within paraxial space Eq. (1.40) appears as follows

1 kkk , (1.41) where kkk vv /1 is the relative rcfraction coefficient. Then the path of the refracted ray is described as follows

111 )1( kkkkkk h , (1.42) where ρ is the curvature of the refracting surface.

The refraction termed in matrix form

* Since ( oa , n ) = cos α, the critical angle is determined by sin αcr =v2/v1. Consequently, the total internal

reflection takes place when a ray travels from the higher density medium to the lower density medium (v1>v2).

25

1

1

)1(101

k

k

kkkk

k hh

. (1.43)

After refraction the ray transfer from the (k-1)-th to the k-th surface can be expressed by the following formula (see Fig. 1.8 b)

kkkk Thh 1 , (1.44) where Tk is the geometrical distance along the ray path. We can write Eq. (1.40) in matrix form

1

1

101

k

kk

k

k hTh

. (1.45)

Now, when calculating the ray path for a set of surfaces, we consider the process to be a sequence of matrix products [6].

Reflection and Refraction Problems Problem I. Let the directions of incident and reflected rays be known. A

normal at the point of incidence has to be determined. The task arises when we want to construct a profile of reflector knowing its angle function (ray-tracing function).

Multiplying scalarly Eq. (1.31) by a , we find

2/10

0 2ˆ,ˆ1ˆ,ˆ

aana .

Solving Eq. (1.31) with respect to n , we obtain

2/100 ]ˆ,ˆ1[2ˆˆ aaaan . (1.46)

In case of refraction, solution of similar problem can be obtained by using Eq. (1.39) [see Fig. 1.9]. Unlike for reflection, the solution here exists not for all a and

oa . Problem 2. A ray is sequentially

reflected by two flat mirrors with the normals n1 and n2, and a common bound m. We have to find out how its direction changes after two reflections.

Fig. 1.9 Determining normal vector to refracting surface with the aid of

cyclotomic diagram 0ˆˆˆ aannk , where

k, n are constants.

26

After the first reflection a ray travels at the direction 111 nkaa

where k1 is a constant. Now a1 becomes an incident ray with respect to the second mirror, and the second reflected ray is

2211 nknkab where k2 is a constant.

Since bm=am, an angle between a ray and a bound remains unchangeable. This holds true for arbitrary number of reflecting and refracting flat bounds. Note, that similar result can be obtained by multiplying matrices from Eq. (1.37).

Problem 3. A light ray traveling at the

direction I1 is being refracted when it strikes a glass with refractive index n=l/q (Fig. 1.10). Having passed the glass, it has to be partly reflected by a mirror bound and to exit through the refracting surface along the direction 2I . The task is to find the orientation of a mirror bound nr if the normal n1 of a front surface is prescribed.

After refraction an incident ray turns to .ˆˆ

1121 nkIqJ After reflection it obtains the direction

.ˆˆ1222 nkIqJ

For the normal nr, we obtain )ˆˆ( 213 JJknr

Parameters k1 and k2 are known, while constant k3 can be found from Eq. 1.46.

Fig. 1.10. Determining normal vector to mirror bound in optical wedge.

27

1.4. RAY AND SURFACE INTERSECTION The task of determining the point of intersection between a ray and a

surface is included as a separate geometric block in the programs package for computing light fixtures. The task has to be solved in two cases: (1) when defining the point, where the ray is incident to a radiance-transforming surface (refracting or reflecting), and finding the transformed ray; (2) when searching for a point where the ray strikes a stretched light source (in the programs based on inverse-ray tracing).

Without loss of generality, while considering these tasks, we may reduce ourselves to observation of surfaces defined by equations of not higher than the second order.

The second order surface is described by a quadratic form. By choosing the coordinate system, the latter can be translated to the diagonal form [1], The surface equation is as follows [7]

1222 zpypxp zyx , or in the matrix form

1RSST , (1.47) where s=[xyz]T, and R is a diagonal matrix

z

y

x

R

0

0.

Let us define an equation for the ray originating from the point T

ooo zyxs ][0 at the direction Tzyx gggg ][ˆ .

lgss ˆ0 , (1.48) where l is the ray length counted from the point so.

Substituting Eq. (1.48) into Eq. (1.47), we get a quadratic equation in unknown parameter l

022 CBlAl . (1.49) The coefficients of equation are [8]

.1

;ˆ

;ˆˆ

20

20

2000

0000

222

zpypxpRsC

zgpygpxgpgRsB

gpgpgpgRgA

zyxsT

zzyyxxT

zzyyxxT

(1.50)

28

We find a solution of quadratic equation, using the following formula ADBl /)( , (1.51)

where D is the discriminant in Eq. (1.49); D=B2-AC. Let the ray start at the point (x0, y0, z0). Then the coordinates of a point

where it intersects the surface arc specified by

.lg,lg,lg

0

0

0

z

y

x

zzyyxx

Arithmetic condition for existence of intersection. In most cases this checking is sufficient to continue program computations, e. g. when calculating luminous intensity of a fixture.

The sign in Eq. (1.51) has to be chosen. We consider two cases: (1) an outside intersection of a surface; (2) an inside intersection (Fig. 1.11). It is clear that 0ˆ ngT in the first case, and 0ˆ ngT in the second case, where n is the normal vector at the point of intersection. We get the normal vector by differentiating Eq. (1.47)

n = Rs . (1.53) Substituting Eq. (1.48) into Eq. (1.53), wc obtain the scalar product

.ˆglˆ)ˆ(ˆˆ 00 lABgRRsglgsRgng TTTT (1.54) Substituting now the expression for l from Eq. (1.51) into Eq. (1.54), we

have DngT .

We have to choose the positive sign near the square root in Eq. (1.51) for inside intersection (see Fig. 1.11a), and the negative

Fig.1.11. Intersection of ray with surface: a - from outside, 0)ˆ,ˆ( ng ; b - from inside, 0)ˆ,ˆ( ng

29

sign in the opposite case (Fig. 1.11b). We find now the coefficients A, B, C and the normal vector for a surface

of revolution having the axis OZ. Wc denote ρ0=ρx=ρy, ρ1=ρz. Then the expressions arc reduced to

.1)(

;)(

;)(

201

20

200

01000

21

220

zyxC

zgygxgBgggA

zyx

zyx

(1.55)

The value of pn in Eq. (1.55) has a meaning of a square of curvature in the section

200 /1,0 Rz , where Ro is the radius of

curvature. We define conic sections in terms of

semiaxes: the minor b and the major a, while 2

0 /1 b , 21 /1 a (Fig. 1.12). The matrix R

has a following form

2

2

01

01

qbR ,

where 22 )/( abq . We take the parameters a and e (e is the

eccentricity) as basic: ).1(,1 22222 eabeq The condition

e<1 represents an ellipsoid; e>1 and b2<0, q2<0, a hyperboloid; e=1, a paraboloid. The coefficients A, B, and C, now take the form

.

;

;)1(1

2220

20

20

2

22

bqzyxC

qzgygxgBgqA

ozoyox

z

(1.56)

We find the normal at the point of intersection s=[xyz]T

.0

101

2

2

2

2

zqyx

bzyx

qbn

Fig. 1.12. Sections of conic surfaces: (e<1 - ellips,

e=1 - parabola, e>1 - hyperbola; a, b are semiaxes;

p is focal parameter; F1, F2 are foci)

30

The length of vector n .)( 2/142222 qzyxbn

The unit normal vector T

k MzqMyMxn ]///[ˆ , (1.57)

where .;;)( 222/12222 qqbbzqyxM kkk

We examine now some specific cases. Cylinder Let a=∞, b=Ro, q=0. The coefficients A, B, C take simple form

.

;;1

20

20

20

0

2

RyxC

ygxgBgA

oyx

z

(1.58)

Obviously, if the point Tooo zyxs ][0 belongs to cylinder, then Eq. (1.49)

turns to linear which has a nontrivial solution ./2 ABl (1.59)

The normal vector at the point of intersection TRyRxn ]0//[ˆ 00 . (1.60)

Sphere Let b=Ro, q=1. From Eq.(1.56) we get

.

;;1

20

20

20

20

00

RzyxC

zqyqxqBA

zyox

(1.61)

The normal vector is directed along the radius towards the point s TRzRyRxn ]///[ˆ 000 .

31

Cone A cone may be considered as an

extreme of a hyperboloid under 0,0 ba , while qk /1tan .

Obviously, the focal length c=0 (Fig.1.13). Thus, we get

.

;;)1(1

20

20

20

00

2

k

kzyox

zk

qzyxC

qzqyqxqBqqA

(1.62)

The unit normal vector n [sec Eq. (1.57)]

TMzMyMxn ]///[ˆ 000 . (1.63)

Paraboloid A paraboloid is a particular case of ellipsoid when the second focus tends

to infinity while focal parameter p=b2/a=const. Let e=1-ε, where ε is a small given value, then 2c and b should be chosen as follows

;//22 2 pqepc (1.64)

.2// pqpb (1.65) We relate deviation ε with the divergence of the rays originated from the

focus and reflected by the surface (quasi-paraboloid). The divergence of reflected axial rays can be determined by differentiating the equation for conic section termed in polar coordinates

cos1 epr

,

that yeilds

.cos21sin2

2eededa

Having calculated the derivative at the point e=1, we get

,2

tandeda

Fig. 1.13. Cone as limiting case of hyperboloid (a→0, b→0)

32

or

.2

tan ea (1.66)

Hence, if we take e=0,995, then for φ=90° we obtain the divergence of reflected ray from the axis ∆φ=0,3°.

Finding the Point of Intersection with a Plane In this case the surface is defined by linear form. The equation for a plane

being in parallel to the plane XOY and situated at the distance H from z = 0, is

HksT ˆ . (1.67)

Substituting Eq. (1.48) into Eq. (1.67), we have

Hkks TT ˆglˆ , whence,

zT

To

gzH

kgksHl 0

ˆˆ

ˆ

. (1.68)

Substituting the expression for l into Eq. (1.52), we find the coordinates of the intersection point.

1.5. RAY TRACING FOR DISCRETE-POINT

REPRESENTATION OF A SURFACE Evidently, we may represent a surface of revolution in the following form

0)( 22 zyxf , (1.69) or in vector terms

0ˆ))(( kssuf T , (1.70)

where, as usual, s is the vector of a surface point and 22 yxu . Solving jointly Eqs. (1.49) and (1.70), we get an equation in l

0ˆ)ˆ())ˆ(( 00 klgslgsuf T . (1.71)

33

Let a generatrix curve f (x) have a discrete-poinl assignment, i. e. it is defined by arrays mixz ii ,1,, . To solve Eq. (1.71), we must know description of a function f (x) at an arbitrary interval 1,1],1,[ mixx ii ,. Thus, at any interval 1,1],1,[ mixx ii the curve is substituted by its

representation ff where

,)( ii zxf ,1)1( ii zxf 1,1 mi . (1.72) Interpolation in terms of Lagrange’s polinomial Lm-1(x) led through m

points is a usual method for analytical representation of discrete function. But when the power is growing the Lagrange’s polinomial may perform strong oscillations between the nodes (Runge’s effect), that leads to unwanted results in ray-tracc calculations. To avoid oscillations, piecewise smooth interpolation by polinomials of low power (not higher than the third power) is applied. Interpolation by segments of a parabola (Ermolinskii’s method) and by segments of a circle (Boldyrev’s method) is widely used. The disadvantage of these methods lies in unsmoothness of curves and their derivatives at the connection points. Interpolation by cubic splines is efficient and widely used method [9, 10] that provides a conjugation between curves and derivatives up to the second order, i. e. to the coincidence of curvatures at the gluing points. Thus, within an arbitrary interval [xi, xi+1] we have representation

)()()( xqf ii , (1.73) where

32)( )()()()( iiiiiiii xxdxxcxxbzxq , (1.74)

while splines )()( xq i meet the condition in Eq. (1.72). Consequently, we have a solution of the equation [see Eq. (1.71)]

miklqslqsuq Ti ,1,0ˆ)ˆ())ˆ(( 00)( . (1.75)

Obviously, the left side of it depends on the parameter l in a complex way, that is why the equation has to be solved by the step-by-step approximation method.

Annex 1 contains the computing program, which is a C-realization of Forsythe’s ZEROIN program [11]. The subroutine-function, which presents an array of parameters, can be included in the formal parameters list of a new program version, in contrast to that of FORTRAN program. It is reached by assigning the ZEROIN-type MATH_EXT.H structure and the description of FUN_ZEROIN-type. VU-\ H

34

The function ZEROIN uses the formal parameter of ZEROIN-type, which is a pointer onto a working function and a parameter vector being applied. In this way, the explicit data transfer from the main function to the working one bypassing the global level is provided.

Generally speaking, the cubic spline representation of a specular surface is sufficient for calculating the light distribution. Further increase in profile-curve smoothness, for example, when persistence of the third derivative is required, docs not improve the smoothness in luminous intensity curves. Really, as it is shown in Ch. 2, the normal illuminance in an infinite pcncil-bcam section at an arbitrary distance from the mirror point is fully dependent on the normal vector direction and the principal curvatures at the mentioned point, in other words on the first and second derivatives.

The calculation of the normal at the intersection point is based on the following considerations. We differentiate the surfacc equation (1.70) and obtain

0ˆ

k

su

uF

sF

. (1.76)

Translating Eq. (1.76), we find the normal vector in the following form T

uu uxf

uxfn

1 ,

where, as it was earlier, 22 yxu , and we denote uffu / . After normalization of vector n, we have

T

uu

u

u

u

fuy

ff

ux

ffn

2/122/122/12 )1(1

)1()1(ˆ . (1.77)

The function f (x) is described by a set of splines, it is twice-differentiated within the interval [x1,xm], therefore, for an arbitrary u there exists a representation

2)(3)(2)(` iiiiiiu uuduucbuqf . (1.78) Calculation of the first and second derivatives may be carried out by the

program SEVAL if we insert the following operators [11]

).(**.3)(*.2));(**.3)(*.2(*)(

IDDXICSPPIDDXICDXIBSP

In the C-version of the program SEVAL the function value and

35

the first and second derivatives are being returned. The pointer to the structure containing the arrays of interpolated function, the coefficients of interpolating function, and the type of interpolation (linear or spline) are being transmitted through the heading as well [see Annex 2]. The reflected ray is determined according to the law of specular reflection [see Eq. (1.31) ]

ggnns ˆ)ˆ,ˆ(ˆ2ˆ , where n is calculated in accordance with Eq.(1.78)

1.6. SURFACE SYNTHESIS BY USING FERMAT’S PRINCIPLE The Fermat’s principle enables to enlighten focusing and concentrating

properties of a surface most completely [15]. It is a known fact that for a path of any light ray passing between two fixed points A and B the following relationship is true

0B

A

ds , (1.79)

where )(s is the refractive index related to the path. For a discrete series of refracting surfaces, when the refractive index is

approximated by a pieccwisc linear function, we obtain 0

kkk s , (1.80)

where all the surfaces are included in the sum. Let the rays being radiated from the point F1 converge at the point F2 (Fig. 1.14). We calculate the difference between the optical pathlengths of two close rays, and assume that the rays travel in the air before and after reflection. Comparing the ray path F1P and F1Q (Fig. 1.14), we obtain

)()( rrL

0)( r

Fig. 1.14. Optical paths of two close rays converging into point (triangles PQR and PQT

are congruent)

36

and this yields r . This equation unambiguously defines a curve (cllips) that collects rays at

its real focus aconstr 2

From congruence of triangles PQR and PQT we obtain (Fig. 1.14) sin i = sin r (the law of reflection),

and rPQiPQ coscos .

The latter equality yields r (1.81)

If the rays arc gathered at a point of imaginary focus, this results in equation

0r , whence, aconstr 2 , i. e. hyperbola equation. Hence, r , whether we consider elliptic or hyperbolic surfaces.

Generally, dealing with a collection of reflecting surfaces, we have

0k

k , (1.82)

where all surfaces are included in the sum, and the sign choice is made in accordance with the said-above remark.

Similarly, using Eq. (1.82) for a number of reflectors, we find

0k

kkr , (1.83)

where we choose + for hyperbolic, and - for elliptic surfaces.

Equation (1.83) being taken with a sign + can be applied to a ray tracing through a series of refracting surfaces.

We now give the examples of designing (synthesis) optical surfaces by applying Eqs. (1.82) and (1.83).

37

Parabolic mirror Let a surface be required that

transforms the flux of a point light source into a beam of parallel rays (Fig. 1.15). According to Fig. 1.15, we have dzrd . Knowing that sinrz , we obtain a differential equation with separable variables

drd )2/tan()(ln . Integrating the equation under the

initial condition of fr )0( , we obtain

cos12

fr (1.84)

Spherical lens We consider a spherical refracting surface which separates the mediums

with refractive indexes η1 and η2 (Fig. 1.16). We find the condition under which the paraxial rays passing from the first medium into the second one arc being focused.

According to Fermat’s principle the optical pathlengths are being equal for all the rays traveling from the point O to the point O' [16]. An optical pathlength over-run for an arbitrary ray is 2211 rr , and it has to be equal to an optical length surplus of a central ray, that is x )( 12 .

Let a height where a ray intersects a surface be h=1. Supposing that the rays are close to the axis, we obtain

122 2))(( rldldldll .

Thus,

Fig.1.16 Spherical lens

Fig. 1.15. Optical properties of parabola in differential form

38

lr 2/11 , `2/1 lr , Rx 2/1 , where R is the radius of a surface. We obtain the focusing condition as follows

Rv

lv

l1

`1

or

flv

l1

`1

(1.85)

where f=R/(v-1) is the lens focal length, and 12 /v is the relative refractive index.

We have to note, that the signs rule accepted in optics [13] is used in Eq. (1.85).

Taking v=1 in the second term in Eq. (1.85), we gel the well-known lens formula.

Spherical mirror Setting v=-1 in Eq. (1.85), we get an equation valid for spherical mirror

flv

l2

`1

(1.86)

A linear extension of an image is determined by relation (1/l), and an area extension is a square of linear one, or

2))2/(( RlRM (1.87) Hence, a concave mirror (R>0) amplifies luminous intensity of a light

source [17]. For the curvature radius of R = 21 the amplification is infinite, and this is a property of a parabolic reflcctor [see the formula for meridian section curvature in Eq. (1.26) when u=0].

It is clear, that occurrencc of infinity at caustic point can be explained by approximate nature of the optics. Using the laws of geometrical optics, we assume that the strength gradient of electromagnetic field in considered region is small in comparison with the field itself [13|. Obviously, this condition is violated near focal points or caustics.

Though being restricted to some extent, the approach in terms of geometrical optics is often useful because of its simplicity and inversion-ability of formulas (transfer from direct to inverse

39

problems). For example, an elementary analysis using Eq. (1.87) shows that uniformly spreaded defects of a flat mirror in the form of small concavities and convexities cause the sharpening of luminous intensity curve, and this fact is confirmed by photometric data [171.

Hyperbolic lens A lens is required which, having

the refractive index v, transforms a flux of a point source in the air into a beam of parallel rays (Fig. 1.17). Examining Fig. 1.17, we obtain a differential equation dr=v dx. Accounting that x=r sin φ and separating the variables, we get

d

vvrd

1cossin)(ln

,

from where, integrating under the condition v(0)=f we have

1cos)1(

vvfr (1.88)

i. e. for v>1 we get a hyperbola equation. When a lens which transforms the rays traveling from a medium with

higher density to a medium with lower density is required, we have to assume v→1/v in above equation. Obviously, we get a lens with elliptic profile [15].

To find an equation of reflecting surface, we have to assume v=-1 in Eq. (1.88), and then we get a parabola equation (1.84).

Double-surfaced reflector (Cassegrain and Gregory’s schemes) After two reflections the light rays run in parallel to the axis (Fig. 1.18).

The second surface is assumed to be parabolic. Accounting Eq. (1.83), we write down dzdrdr 2211 ` . We assume that the first surface and the parabolic surface have a common centre, so that dz=ρ dθ=p dθ2. Hence, we define an equation for the first surface 2211 drdr , and this is the equation of conic section [15]. Precisely, the upper sign (+) defines a hyperbola

Fig. 1.17. Hyperbolic lens

40

in Cassegrain’s scheme; and the lower (-), an ellips in Gregory’s scheme.

One important thing has to be mentioned. The obtained-above curves, for instance, described by conic section equations, arc the only curves that fit the stated requirements for rays transformation. It follows from the fact, that they have been resulted as solutions of ordinary differential equations under given initial conditions.

The conditions of Cauchy’s theorem are met here, so these solutions are unique.

Ray-tracing function In contrast with the previous examples, we have a surface specified by a

profile curve r=r(φ), and the task is to reconstruct a relationship between corresponding incident and reflected rays (ray-tracing function) α=α(φ). We determine the function α=α(φ) for conic sections assigned in polar terms

)cos1/( epr . We take a derivative )cos1/(sin)/( eedrdr . Setting it equal to the right side of Boldyrev’s equation

)2/)tan((/ drdr ,. (1.89) we obtain

)2/(cos2cos1)2/(cos2cos1)2/tan()2/tan( 2

2

eeee

and finally,

)2/tan(11)2/tan(

ee

(1.90)

From Eq. (1.90) it follows that angles a have to be assumed as positive, i. e. in case of ellips (e<1) the rays arc being directed towards the optical axis; and as negative in ease of hyperbola (e>l), when the rays are being directed from the optical axis. Hence, the double sign in Eq. (1.90) is not necessary.

Fig. 1.18. Scheme with double reflection

41

Conic sections formulas For further convenience we give the formulas for conic sections in

different coordinate systems. (1) Cartesian system. The coordinate origin is situated at a symmetry

centre of a curve (canonical representation)

12

2

2

2

by

ax

. (1.91)

(2) The origin is matched with a vertex of a curve 222 *)1(2 xepxy . (1.92)

(3) Polar system. The pole is matched with a focus of a curve

cos1 epr

. (1.93)

The parameters in given formulas have the following meanings: a and b are half-axes, a>b;

,,222 bcbac where c is the distance between the focus and the centre; ace / is the eccentricity; )1(/ 22 eaabp is the parameter of a conic section.

Remarks (1) The sign (+) in Eq. (1.91) corresponds to ellips; (-), to hyperbola; (2) Parabola is specified by parameter p or by focal length f=p/2. Cofocal conic sections are defined by the following formula

12

2

2

2

b

ya

x, (1.94)

where λ is a family parameter. It follows from the definition c2=(a2+λ)-(b2 + λ)=a2-b2 that curves have one and the same focus.

42

1.7. MATRIX APPLICATION IN GEOMETRIC AND OPTICAL CALCULATIONS We consider now how matrices arc used in lighting calculations. Spherical Coordinates of a Point The position of a point in space or on a surface is often described by

spherical coordinates: the radius-vector r, the polar angle φ, and the azimuth angle ψ (Fig. 1.19). A vector p coincident by direction with a vector OP of the length r can be termed as a result of two rotations: (1) by the angle ψ in the plane XOY; (2) by the angle ψ in the plane ZOY (Fig. 1.19). To determine matrices of rotation we have to find the coordinates of the revolved basis vectors, which form the columns in the matrix of rotation.*

For the projections of basis vectors 21 ˆ,ˆ ee and 3e on the plane XOY (Fig.1.19) we obtain

.]100[ˆ;]0cossin[ˆ;]0sincos[ˆ

3

2

1

T

T

T

eee

Consequently, the matrix of rotation by the angle ψ is

1000cossin0sincos

M . (1.95)

For the projections of basis vectors 21 ˆ,ˆ ee , 3e on the plane ZOY (Fig. 1.19) wc

obtain

.]cossin0[ˆ;]sincos0[ˆ;]001[ˆ

3

2

1

T

T

T

eee

Consequently, the matrix of rotation by the angle φ is

* We mean the left multiplication of matrices.

Fig. 1.19. Coordinates of point P in space and rotation of basis by angle ψ

43

cossin0sincos0

001M . (1.96)

The resulting matrix is MMM . According to the rule of multiplication of matrices, we obtain

cossin0cossincoscossinsinsinsincoscos

M . (1.97)

Note, that MMMM . Multiplying the matrix of rotation M by

the vector Tk ]100[ˆ , we find the vector p : Tp ]coscossinsinsin[ˆ ,

hence, as prOP ˆ , TrrrOP ]coscossinsinsin[

or .cos,cossin,sinsin rzryrx (1.98)

Rotation of Lighting Fixture Axis Let a lighting fixture (LF), considered as a point source, be placed at the

coordinates origin with its axis directed along the axis OZ. It is usually photometrically tested at a plane, which is perpendicular to the LF axis. Wc take the photometric plane being at a distance z=1.

We assume that the LF is aimed at a point P(x, y) on the plane z=H (Fig.1.20). To find the LF luminous intensity at the direction rs, we do the following: (1) determine the angles of rotation φ and ψ from Eq. (1.98); (2) apply the matrix of inverse rotation 1

M to the vector rs, and

consider the coordinates of obtained (unrevolved) vector on the plane z=1.

Fig. 1.20. Determination of ray coordinates on projection plane

44

Since matrix M is orthogonal and unitary [1], its inverse matrix is equal

to the transposed one, i.e. to TM . Hence,

coscossinsinsinsincoscossincos0sincos

1M . (1.99)

After rotation defined by matrix 1

M , the vector rs, (Fig. 1.20) turns to

sB rMr 1

or

TT HYXMhyx ][]``[ 1 . (1.100)

The coordinates of vector Br in the plane h=1 can be obtained from Eqs.(1.99) and (1.100) in the following form [18]

33*

23*

21

13*

12*

11

mYmXmmYmXmx

; (1.101)

33*

23*

13

23*

22*

21

mYmXmmYmXmy

,

where X*=X/H; Y*=Y/H; mij are the matrix elements. The vector coordinates in the plane h=1 may be regarded as the projection

coordinates of a point. The formulas (1.101) describe the most general view of coordinate translation in an adduced plane. In particular case, when ψ=0 and φ→900-φ, Eq.(1.101) transposes into the known expression from Sapozhnikov’s method [19].

If the luminous intensity is prescribed in the plane h=1 as a matrix Jij i. e. discretely with spacings of hx and hy, then the candlepower related to the LF aiming point can be determined by reading from the array J(i,j) with i=[x/hx]+ 1, j=[y/hy]+1, where expression being put in square brackets denotes an integer part of a number.

Earlier we saw that the matrix of reflection takes the most simple form being represented in the basis, where one of the vectors is directed along a surface normal at a point of ray incidence. This is because the basis consists of eigenvectors of reflection matrix Mrefl from Eq. (1.34). Defining the normal at a point of incidence in terms of spherical coordinates, i. e. by the angles δ and ψ (Fig. 1.21), we obtain the expression similar to Eq. (1.98)

45

Tn ]coscossinsin[sinˆ . (1.102) The vector n described by Eq. (1.102) can be resulted from the translation

of the basis axis vector OZ ( k ) into the vector n . Consequently we pass to another coordinate system, to a new vector basis 21 ˆ,ˆ ee , 3e e.i, where 3e is directed along the normal; 2e , along the meridian ψ=const; and 1e , along the parallel φ=const (Fig.1.21).* If a reflecting surface is produced by rotation of a curve r=r(φ) about the axis OZ, then )( , and the parallel is defined by the line φ=const. Obviously, the coordinates of an arbitrary vector within the basis 21 ˆ,ˆ ee , 3e are determined by applying inverse matrix 1

M similar to that

from Eq. (1.99)

coscossinsinsinsincoscossincos0sincos

1M . (1.103)

Let an incident ray at point P be set as Tozoyoxo ssss ][ˆ . In the local

system of coordinates it is denoted as vector s`o

oo sMs ˆ`ˆ 1 , (1.104)

where matrix 1M is defined by Eq. (1.103).

After reflection, the ray travels at the direction assigned by vector s [see Eq. (1.36)]

Fig.1.21 Local coordinate system at mirror point

* Vectors 21 ˆ,ˆ ee , 3e compose the right triple.

46

Tozoyox ssss ]```[`ˆ , (1.105)

where ozoyox sss `,`,` are coordinates of vector os`ˆ determined from Eq. (1.104).

The matrix from Eq, (1.103) enables to solve all the tasks connected with reflection from a specular surface. Let an incident ray be set as T

os ]100[ˆ . Then according to the rule of matrix multiplication, this ray can be defined in the local coordinate system

TTo Ms ]cossin0[]100[` 1 .

Obviously, the reflected ray is Ts ]cossin0[`ˆ .

In the main coordinate system the reflected ray is defined as TsMs ]2coscos2sinsin2sin[`ˆˆ .

We have got the result well-known in optics; when a normal is being rotated by angle δ, reflected ray rotates by the double angle 2δ . The formulas (1.103)-(1.105) are suitable for automated calculations of ray-paths.

When transferring to the local coordinate system, some additional considerations have to be accounted. For example, within inverse ray-tracing method when searching for an intersection point between a ray and a light source, it is more convenient to translate a ray into the coordinate system associated with a light source, thus presenting a light source surface in the most simple form.

Coordinate System Associated With a Light Source Let a light source be rotationally

symmetric. It is convenient to direct the basis vector, say 3e , along the axis of symmetry (Fig, 1.22).

As it follows from said above [see Eq. (1.99)], in order to define the matrix of rotation it is sufficient to find the coor-dinates of novel basis vectors within original coordinate system XOZ associated with reflector; these coordinates give the columns of desired matrix.

Fig.1.22. Displacement of coordinate system into point O' associated with light source

47

Let Twnue ][ˆ3 , where 1222 wvu . In the absence of other

considerations it is suitable to choose Tuve ]0[1 . It is easy to prove that

1e is orthogonal to 3e . Vector 2e is orthogonal to 1e and 3e as well. Therefore,

2e can be specified by vector product between 3e and 1e ]ˆˆ[ˆ 312 eee ,

or

0

ˆˆˆ

ˆ2

uvwvukji

e

.

Expanding the determinant, we get Twvwuwe ]1[ˆ 2

2 . Hence, the matrix of transfer to the coordinate system associated with a

light source takes the form

wwvvwuuuwv

M s

10 2

. (1.106)

Let a point P be assigned by vector Rp=[xp yp zp]T in the coordinate system associated with reflector (the main system). The origin of the local coordinate system is Tzyxr ][ 0000 . Then in the local coordinate system this point is defined by equations

Tppppsp zzyyxxrRr ][ 0000 ;

spssp rMr ` , where Ms is calculated from Eq. (1.106).

Light Source Image Produced by Reflecting Surface

Reflection is a linear transformation, as it is shown in paragraph 1.3. Consequently, if a and b are arbitrary vectors, while p and q are scalar constants, then

bqMapMqbpaM reflreflrefl )( . (1.107) Since every vector can be defined within the basis, then, according to Eq.

(1.107), its transformation by a specular reflector can be found. Let a surface of revolution be assigned by angles φ and ψ. The basis of

rectangular coordinate system is specified by vectors

48

Fig. 1.23. Image of light source produced bv element of specular surface: a - orientation of source, mirror, and pictorial plane;

b - images of basis vectors

21 ˆ,ˆ ee , 3e (Fig. 1.23). We direct the normal being al the point φ, ψ outwards the surface; the angle between the normal and the axis OZ is δ (see Fig. 1.21). Substituting the coordinates of vector n into Eq. (1.34), we obtain the matrix of reflection in the following form

1cos2cos2sinsin2sincos2sin1cossin2sinsinsinsinsinsin1sinsin2

2

2222

22222

reflM . (1.108)

We find now the images of vectors et and 1e and 2e

;sinsinsinsin1sinsin2ˆ` 2222211

TrefleMe (1.109)

.cos2sin1cossin2sinsinˆ` 222222

TrefleMe (1.110)

For the points )0( hP and )90( 0hP [see Fig. 1.231 the images of vectors 1e and 2e on the plane being parallel to XOY (the plane of observation or pictorial plane) take the form

;90,]02cos[;0,]01[

`01

T

T

e (1.111)

;90,]10[;0,]2cos0[

`02

T

T

e (1.112)

Thus, according to Eqs. (1.111) and (1.112), vectors may change orientation, and their length may be subjected to projective shortening. Let us find the angle of rotation of vector 1e after

49

reflection (Fig. 1.23 b). Taking the coordinates of 1e from Eq. (1.109), we obtain

1cossin2sinsin

``

tan 22

22

1

1

x

y

ee

. (1.113)

Transformations in Eq. (1.113) lead to

2costan1tan)2cos1(tan 2

. (1.114)

The maximal angle of rotation can be obtained by differentiating Eq. (1.114)

2cos1tan max . (1.115)

Equations (1.109) and (1.110) describe rotation of elementary map of a linear source being perpendicular to the optical axis of reflector with arbitrary geometry. They are similar to that of paraboloid, if we take 2/ [20].

For reflector points specified discretely with steps hφ and hψ, the matrix of reflection [Eq. (1.108)] can be determined in advance. The use of matrix multiplication [see Eq. (1.35)] enables to create effective algorithm for determining light source images on an arbitrary pictorial plane.

For a light source, in particular, having rectangular form, wc may delete the restriction on its dimensions l and h [20] (they supposed to be small enough). Any contour point of rectangle may be assigned by vector

21 ˆˆ eets , where 10 t ; h 0 ; t and τ are some scalar constants.

In accordance with Eqs. (1.107), (1.109), and (1.110), we obtain full description of elementary maps for the points Ph and Pv that belong to a paraboloid

.,cos:

,cos,:

yxr

yxh

ssPstsP

(1.116)

Note, that in the course of linear mapping described by matrix Mrefl, straight lines transform into straight lines, and a trace of elementary image on a plane is rectangular as well. If a grid of test points is specified on the pictorial plane, then it is easy to develop an algorithm that makes record when an elementary image covers a given test point (a node).

50

REFERENCES (1) Streng G. Linear Algebra and Its Applications. Academic Press, New York -

San Francisco - London, 1976. (2) Rice J.R. Matrix Computations and Mathematical Software. Mc Graw Hill

Book Comp., St.Louis - San Francisco, 1981. (3) Rogers D.F., Adams J.A. Mathematical Elements of Computer Graphics. Mc

Graw Hill Book Comp., St.Louis - San Francisco, 1980. (4) Faux I.D., Pratt M.J. Computational Geometry for Design and Manufacture.

Ellis Horwood Ltd., 1979. (5) O’Naill E.L. Introduction to Statistical Optics. Addison-Wesley Publ.Comp.,

Reading (Massachusetts) - London, 1963. (6) Gerrard A., Burch J.M. Introduction to Matrix Methods in Optics. Wiley -

Interscience Publ., London - New York - Sydney - Toronto, 1978. (7) Korn G.A., Korn T.M. Mathematical Handbook for Scientists and Engineers.

2nd edition. McGraw-Hill Book Comp., 1968. (8) Rodionov S.A. Computer-Aided Design of Optical Systems. L.:

Mashinostroenie, 1982 (in Russian). (9) Alberg J.H., Nilson E.N., and Walsh J.L. The Theory of Splines and Their

Applications. Academic Press, New York - London, 1967. (10) Stockmar A. Spline-Funktionen-eine neue Methode zur Darsiellung von

Lichtstarkeverteilungen.// Lichttechnik, 1975, N8, s.320-324. (11) Forsythe G.E., Malcolm M.A., Moler C.B. Computer Methods for

Mathematical Computations. Prentice-Hall, N.J. 1977. (12) Kusch O.K., Sofronov N.N. Computer Calculation of Specular Reflectors by

Using Splines.// Svetotekhnika, 1985, N4, p. 12-13 (in Russian). (13) Born M., Wolf E. Principles of Optics, 4th cd., Pergamon Press, Oxford,

1970. (14) Stavroudis O.N. The Optics of Rays, Wavefronts, and Caustics. Academic

Press, New York - London, 1976. (15) Cornbleet S. Microwave Optics. Academic Press. New York - San Francisco

- London, 1976. (16) Feynman R.P., Leighton R.B., Sands M. The Feynman Lectures on Physics.

Vol. 1. Addison-Wesley Publ. Comp., Massachusetts, Palo Alto, London, 1963. (17) Benford F. Studies in the Projection of Light. General Electric Review, 1923-

26. (18) Vakhromeeva L.A., Bugaevskii L.M., Kazakova Z.L. Mathematical Carto-

graphy. M. Nedra, 1986. (in Russian). (19) Knorring G.M. Lighting Calculations in Installations of Artificial

Illumination. M.-L., Energiya, 1973. (in Russian). (20) Karyakin N.A. Lighting Devices of Floodlighting and Projection Types, M.

Vysshaya Shkola, 1966. (in Russian).

51

CHAPTER TWO METHODS FOR CALCULATING LIGHT DISTRIBUTION 2.1. LIGHT-RAY METHODS Light-ray methods can be originated from two propositions [1]: (1) The property of luminous field to form light-tubes that transmit one

and the same luminous flux through their arbitrary cross-section. This is expressed by the following equation: divE=0;

(2) The existence of a family of orthogonal sections in a light tube (Maluce’s theorem).

We consider a light-tube falling at a mirror surface and cutting an area dvdu there (Fig. 2.1). The mirror-surface element receives a luminous flux

dvdur

rrIIФd vuo2

][ˆ , (2.1)

where Io is the luminous intensity at the tube; r=r(u, v) is the ray equation in curvilinear coordinates u, v related to the surface.

During reflection, a part of luminous flux is being lost. The rest of luminous power spreads within the other light-tube that ends at

Fig.2.1. On energy balance within incident and reflected tubes

52

the target surface by an element ][1 vu RRI du dv, where 1I is the unit vector of reflected ray. Consequently, the normal illuminance at the target surface is [2], [3]:

][ˆ][ˆ

12

0

vu

vuon RRIr

rrIIE

* (2.2)

Expression for the normal illuminance at the target surface can be presented as follows

),( vuDEE on , (2.3)

][ˆ][ˆ

),(1

0

vu

vu

RRIrrIvuD

where ),( vuD is the coefficient equal to the ratio between illuminances in incident and reflected beams; E0 is the illuminance at mirror surface.

Hence, the multiple D(u,v) is the Jacobian of the transformation dS1→dS2 [4].

Now we determine a field in remote zone. A vector corresponding to a point at the target plane is

1IrR . Partial derivatives of vector R are

,ˆ,ˆ11 vvvuuu IrRIrR (2.4)

Substituting Eq. (2.4) into Eq. (2.2), we obtain

)ˆ)(ˆ(ˆ][ˆ

1112

0

vuuu

vuon IrIrIr

rrIIE

. (2.5)

Multiplying Eq. (2.5) by σ and assuming σ→∞, we obtain the value for intensity at the remote field

]ˆˆ[ˆ][ˆ

111

00

vu

vu

IIIrrIEI

. (2.6)

* Since E-value is positive, mixed product should be taken with relevant sign (+) or (-).

53

Formula (2.6) defines the field at the remote zone in terms of proximate field. We set 000 Irr . Substituting r into Eq. (2.5) and supposing σ0→∞, we obtain

]ˆˆ[ˆ]ˆˆ[ˆ

111

00

vu

vu

IIIIIIII

. (2.7)

If the candlepower pattern ),(0 vuI of a point source is known, expressions in Eqs. (2.5)-(2.7) allow to determine the intensity at the remote and proximate fields. Light-tubes directions can be obtained by applying the law of reflection

)ˆ,ˆ(ˆ2ˆˆ001 nInII . (2.8)

Surfaces S1 and S2 in Eqs.(2.2), (2.5), and (2.7) are prescribed in parametric form

,),(),(),(

,),(),(),(vuZvuYvuXR

vuzvuyvuxr

(2.9)

whence, we obtain T

vvvvT

uuuu zyxrzyxr ][,][ T

vvvvT

uuuu ZYXRZYXR ][,][ . (2.10) Equations (2.3), (2.5)-(2.10) make up the closed sequence for calculating

the reflected field intensity when the point-source candlepower distribution is known.

We must note, that similar equations based on geometric representation of luminous field arc being used for calculating scalar density of diffused energy reflected from antennas [5], when designing simulators of thermal radiation [6], and in the wave theory [7]. Spherical coordinates u=φ, v=ψ are often being chosen as reflecting surface parameters. Since surface equation is prescribed, consequently, the pathways equations for reflected rays a=a (φ,ψ) and β=β (φ,ψ) are known as well (where a and β are spherical coordinates of reflected ray). The unit vectors for incident and reflected rays are as follows

T

T

I

I

coscossinsinsinˆ,cossinsinsinsinˆ

1

0

(2.11)

Differentiating Eq. (2.11) and substituting the results into Eq. (2.7),

54

after transformations we obtain an expression known from [8]

a

a

a

D

sin

sin),( (2.12)

The use of obtained expressions for computing the intensity of reflected field is connected with the coordinate representation of vectors and surfaces. When the number of surfaces is large, the other formulas are preferred; the formulas, which present intensity in invariant form, i. e. as a function of geometric and differential properties of reflecting surface, and parameters of incident and reflected rays.

Formulas for calculating coefficient D Table 2.1 Source type Vector form Coordinate form

Parametric representation Spatial source:

proximate

zone

remote zone

][ˆ][ˆ

1

0

vu

vu

RRIrrI

vvv

uuu

zyx

vvv

uuu

ZYXZYXlll

zyxzyxzyx

zyx111

2/1222 :)(

]ˆˆ[ˆ]ˆˆ[ˆ

111

000

vu

vu

IIIIII

vzvyvx

uzuyux

zxx

vvv

uuu

lllllllll

zyxzyxzyx

zyx

111

111

1112/1222 :)(

Linear source:

proximate zone

remote zone

)ˆ,()ˆ,(

1

0

IRIr

u

u yuxu

uu

lYlXyyxxyx

11

2/122 )(

)ˆˆ()ˆ,ˆ(

11

00

IIII

u

u

yyxx

yx

llllyloxlo

yxuu

uu

1111

2/122 )(

Invariant representation Spatial source: proximate zone

remote zone

12 )12( reflrefl HK

120

20 ]~4)]tan~2(cos21[ KkH

Linear source

proximate zone

remote zone

10

1

10

)(cos211

R

1

1

0

cos21

R

Remark. We have E=I0D at the proximate zone, and J=IoD at the remote zone. We take an absolute value of D.

55

Equations for the value F=l/D, known as divergence, were obtained in a general form by V.A. Fock [9], and are as follows*

122 reflrefl HKF , where

20

2 1)tan2(cos2~4

kHKK refl ;

0

2 1)tan2(cos

kHH refl ; (2.13)

21

1~RR

K ,

21

1121~

RRH ,

2

2

1

2 sincosRR

k .

The following denotions are used in Eq.(2.13): K~ , H~ are the Gaussian and mean surface curvatures at the point of ray incidence; R1, R2 are the radii of principal curvatures of a reflecting surface at the point of ray incidence; δ is the angle between the plane, which contains the ray, and the plane of the principal normal section; k is the curvature in the section containing the ray; θ is the angle of ray incidence; σ0 is the distance between the light source and the intersection point; σ is the distance along the reflected ray up to the point at the target surface.

For the remote zone (σ→∞) the divergence takes more simple form [10] KkHF ~4)tan~~2(cos21 2

02

0 . (2.14) The ratio D=F-1=I/I0, where F is calculated from Eq. (2.13), may be

defined as an indicatrix of reflected radiation (or as a direction diagram) in the remote zone. It is an advantage that Eqs. (2.13, 2.14) do not depend on the choice of coordinate system, because all the values here have a particular physical meaning. Given formulas describe reflected field intensity in the case when the light source possesses the spatial intensity distribution of Io(u, v).

Complete summary of D-factor expressions for various radiators, including linear source I0=Io(u), and surfaces representations is given in Table 2.1 (where indices u, v denote partial derivatives with respect to corresponding coordinates).

* Similar equations for calculating the amplification factor of a specular surface are known in lighting engineering [8].

56

Example Let us consider the calculation of the illuminance pattern created by

paraboloid with a light source in its focus. Exact formula for calculating illuminance from paraboloid can be obtained from Eq. (2.13). We have (see Ch. 1) )2/(cos 2

0 f , )2/(cos2 31 fR , )2/(cos2 1

2 fR . Substituting these equalities into Eq. (2.13), we find the divergence to be

1F . Therefore, when I0=1 ang p0=1, it yeilds

2cos11 4

220

f

En . (2.15)

Using transformation φ=2arctan (x/2f), Eq. (2.15) yeilds a function with respect to x-coordinate

222 ])2/(1[1

fxfEn

. (2.16)

According to theory, the value of En does not change when the distance from the optical system increases. It means that the light-tube originated from paraboloid has a beam shape inside of which a constant amount of energy flows. It is seen from Table 2.1 and Eq. (2.13) that in the reflected field the points where F=0 (the caustic points) may appear. In the programs based on ray methods the special means have to be foreseen to prevent computer overflow when caustics occur, and corresponding messages signalling about caustics have to be printed.

Light Distribution Created by Surface with Linear (Filament) Source Let )(00 be the equation for the profile of cylindric surface.

Obviously, the profile curve is identically defined (when the initial point )(0 is prescribed) if the ray-tracing function a=a(φ) is known.

According to the formulas from Table 2.1, we determine the normal illuminance

))cos/(2/1/1( 00 aRI

En

, (2.17)

Supposing differentiability of a(φ), we find the curvature radius of the profile

57

aaR

cos)1(2 0

, (2.18)

where a =da/dφ. Substituting (2.18) into Eq. (2.17), we get

aI

En

0

. (2.19)

We consider some specific cases in defining profile curves: (1) A straight line 02 a

1 ; )/( 0 IEn . (2) A parabola const

0 ; 0/IEn . It follows that parabola as a profile curve can be defined in two equivalent

ways (for I=const): (1) as a curve which originates a parallel beam of light; (2) as a curve which provides the normal illuminance being inversely

proportional to the radius-vector of a section point for cylindric surface, and to the second power of radius-vector for rotationally symmetric surface.

Location of caustics can be found from the following equation 0))cos/(2/1/1( 0 R , (2.20)

or from equation being linear with respect to σ 00 . (2.21)

The above-stated equations can be obtained directly from the Fermat’s principle [see 1.6].

Assuming the angles to be small and cosa=1 (paraxial approximation), we get the equation of focusing [Eq. (1.81)].

Correspondingly, we obtain

aRaR

cos2cos

0

0

, (2.22)

and

a

0 , (2.22)

58

From Eq.(2.23), in particular, it is seen that all the singular points of parabola appear in infinity. The parameters σ, R, cosa depend on surface variable u only. To specify a surface through )( -characteristic, we may set up u=φ.

The caustics surface is described as follows )(ˆ)()()( uauuruR , (2.24)

where r is the radius-vector of reflector point, a is the vector of reflected ray. By tending in Eq. (2.24), we obtain the indicatrix of the reflected

light

II , (2.25)

which is in complete agreement with the flux balance. For point-discrete assignment of a profjle curve the most correct way is to

specify the function )( , because only single differentiation of Eq. (2.25) is needed. This situation occurs when, e. g. the functions )(rr and

)( are obtained as inverse problem solutions, and the direct calculation, has to be carried out.

Similar results can be obtained in gerteral Case, when a surface is specified by a pair of parametric functions ),( vu , ),( vu which describes the paths of reflected rays. Location of caustics is defined by the roots of square equation

01),(),(2),(),( 2 vuvuHvuvuK , (2.26) while the equation of caustic surface is as follows

),(ˆ),(),(),( vuavuvurvuR .

2.2. LIGHT DISTRIBUTION CREATED BY ELLIPSOID OR HYPERBOLOID WITH A POINT SOURCE

Reflectors used for illumination can be divided into two groups:

hyperboloids and ellipsoids. Different kinds of ellipsoids are deep and narrow, they converge the rays towards the optical axis; while hyperboloids are wide and shallow, and diverge the rays outwards the axis.

After passing through the point of secondary focus the beam of ellipsoid appears like that of hyperboloid; so to say, optical equivalence takes place [11]. For rather wide beam, the size of a

59

lighl source effects only on the resulting angular width of a lightspread curve, thus the point-source theory can be used for analysis of ellipsoids and hyperboloids [11, 12].

Returning to Fig. 1.14, we find ar sinsin . (2.27)

Taking into account Eqs. (1.81) and (2.27), we obtain a differential equation that defines a relationship between the angles a and φ (the ray path)

sinsin a

dda

. (2.28)

Light Amplification in Remote Zone Now we determine the amplification factor Kam yielded by a surface.

Taking Eq. (2.12), we obtain for axially symmetric surface

dadaКат /sin

sin ,

or accounting Eq. (2.28) 2)sin/(sin aКат . (2.29)

We calculate the quantity

)cos21()1(

2/tan12

tan111

2tan1 2

2

22

22 ee

eeea

.

Further we find

)cos21)(2/tan1(

)1(2

tan112

2/tan12/tan2sin 22

2

2 ee

eee

aaa

,

or

2

2

cos211sinsin

eeea

.

Hence, we obtain from Eq. (2.29) (sec [14]): 2

2

2

1cos21

eeeKam

. (2.30)

60

Equation (2.30) combined with Eq. (2.22) completely defines luminous intensity distribution of hyperboloid or ellipsoid in the remote zone within the whole beam section except for its boundaries, where edges are not so sharp as it follows from Eq. (2.30).

Equation (2.30) can be represented in equivalent form [11]: 2)/( rpKam . (2.31)

Consequently, for a convex hyperbola with Kam<1 the reflected light beam is more diffused and less intensive than the initial radiation of a light source. The observed picture is inverse to that of a concave hyperboloid: light beam has minimal intensity in the centre and reaches maximum value equal to the light source intensity at the edges, provided that reflector is expanded to infinity [12].

For cylindric reflecting surface with a linear light source we obtain [14], in much the same way as per Eq. (2.30), the following

2

2

1cos21

eeeKam

, (2.32)

where the upper sign refers to elliptical reflectors, and the lower sign, to hyperbolic ones.

Light Amplification in Proximate Zone

To determine the illuminance we

apply Eq. (2.13). Futher, we have to find angle v between the radius-vector and the tangent to a conic section curve (Fig. 2.2).

Differentiating the radius-vectors, we obtain

1

tan

ddrrv .

Since )cos1/( pr , we have sin/)cos1(tan ev ,

Fig. 2.2. On deduction of amplification

factor in proximate zone

61

and futher,

.)cos21(

sin)tan1(

1cos

;)cos21(

cos1)tan1(

tansin

2/122/12

2/122/12

eee

vv

eee

vvv

We find the radius of curvature in the meridian section by applying known formulas (see Ch. 1):

```2)`(

22

2/322

rrrrrrRm

. (2.33)

Now we calculate );cos21()cos1(` 24222 eeeprr

];sin2)cos1(cos[)cos1(`` 2242 eeeeprr ];sin2)coscos21[()cos1(`2 22224222 eeeeprr

3222 )cos1(```2 eprrrr . Substituting the obtained expressions into Eq. (2.33), wc get

vpee

epRm3

3

2/12sin

cos21cos1

.

Applying Meusner’s theorem and setting the radius of a parallel equal to x, we obtain the radius of curvature in the sagittal section

cos/xRs , where is the angle between the normal at the mirror point and the

parallel (Fig. 2.2). We find vv )90()90( 00 ;

cos

)cos21(sinsinsincoscos)cos( 2ee

evvv

2/122/12 )cos21(sinsin

)cos21(cos1

ee

eee

62

Thus, vpeeepRs

112/121 sinsin)cos21(sincos1 . The Gaussian curvature at a point is

24 /sin/1~ pvRRK sm . The mean curvature is

)sin(sin2111

21~ 3 vv

pRRH

sm

.

Applying reflection formulas, we determine

20

2

0

1)tan~2(cos2~4R

kHR

KK refl ,

where 090 v is the angle of incidence; 0;0 rR

,1sincos 22

msm RRRk

vsincos , vcottan .

We find the expression in brackets

pvv

pvvv

pvkH

22

332 sin2cotsin)sin(sin1sin)tan~2(cos

.

Futher, we have 2

2

222222

cos211sin21sin4~41

eeev

prv

prrKrK refl

;

2

222

cos2111sin21)cot~2(sin

eeev

prvkHvrrH refl

The divergence is 22 )1(12 amreflrefl KRHRKF ,

where

Rr

eeeKam

;cos21

12

2

.

63

Thus, the illuminance at arbitrary distance from the focus is

2

2

2

1

cos211

eee

Rr

EEFE nn

, (2.34)

where En is the normal illuminance of a mirror clement produced by a light source.

We consider some specific cases: (1) the candlepower indicatrix or directional diagram in remote zone,

σ→∞;

;1cos21

2

2

22

eeeERJ

(2) the illuminance provided by a paraboloid (e = 1) 20 / rJEE ;

(3) the caustic (we suppose that F= 0 or Kam σ =1)

Rr

eeeKam

2

2

cos211

.

From Eq. (2.23) wc have Kam = r/p, R=p, i.e. the second point is at the focus of elliptic reflector. Since Kam is a nonnegative value, the equality holds true only for e≤1;

(4) spherical reflector (e=0) 22 ]/)[()1( rrRIF ,

that is the intensity deterioration produced by spherical surface (the same result can be obtained from geometrical considerations);

(5) there is a complete equivalence between ellipsoid and hyperboloid in the infinity. At close distance (for the same φ-values) F is greater for e>1, i.e. the beam of hyperboloid is dispersed at a greater state.

64

2.3. INVERSE-RAY METHODS Occurcnce of infinilc illuminance al some field points (caustics), when the

ray formulas (2.13) are applied, is caused by breaking the rules in usage of geometrical terms [10].

Actually, in a fixture with real light source the averaging of luminance over the exit pupil lakes place. Formally, any separate point of the exit pupil can make a definite contribution to illuminance, acting like a δ-source, i.e. a source with infinite luminance.

In accordance with the known expression for the illuminance at the point P, we have

dSRLPE 212 coscos)( , (2.35)

and for the luminous intensity at the direction 0a

dSLaI cos)( 0 , (2.36) where L is the luminance of a flashed area dS of reflector when viewed from the point P along the direction 0a ; R is the distance from the area element dS to the point P; p is the reflectance (further we set up p=1).

Integration in Eqs. (2.35) and (2.36) is carried out over the total reflector surface or over the exit pupil.* The integral here may be treated in sufficiently broad sense, including the δ-function representation of luminance distribution.

We suppose that there exists a system of nonlinear coordinates u, v; i.e. the surface equation takes the form r=r(u,v) or r=[x(u,v) y(n,v) z (a, v)]T. We rewrite Eqs. (2.35) and (2.36) in the following form

;),()ˆ,ˆ)(ˆ,ˆ)(,(1)( 02 dvduvuHnanavuLR

PE pn (2.37)

;),()ˆ,ˆ)(,()ˆ( 00 dvduvuHnavuLaI (2.37) where H(u,v) is the Jacobian of transformation to nonlinear coordinates u, v;

pn is the unit normal vector to the area P; mPR . The vector 0a in Eq. (2.38) is «external» parameter for the integral of luminous intensity, while the viewing vector 0a in Eq. (2.37), generally, is a function of current reflector point P.

On the surface we introduce a grid with cell dimensions hu and

* In accordance with Maxwell’s principle [15].

65

hv along u and v coordinates, respectively

uuoi Nihiuu ,1,)1( ;

.,1,)1( vvoi Nihivv ; For each node we assign the value ksw=1 if a point is perceived as bright,

and ksw= 0 in the opposite case. The discrete analogs of Eqs. (2.37) and (2.38) applicable for computer calculations take the form

;),()ˆ,ˆ)(ˆ,ˆ(),()( 01 1

2 vujipijijoij

Nu

i

Nv

j ij

iisw hhvuHnana

RvuLkPE

(2.39)

;),()ˆ,ˆ(),()ˆ(1 1

0 vujiijoij

Nu

i

Nv

jiisw hhvuHnavuLkaI

(2.40)

Expressions (2.39) and (2.40) give assessments of E and I values with accuracy related to the intervals of discretization hu and hv. These relations arc applicable cither to symmetric or asymmetric surfaces if we possess an independent procedure for calculating the radiance factor ksw at an arbitrary point i, j.

Calculation by using the inverse-ray method [16-20] Realization of the inverse-ray method includes the following stages: (1) description of luminance and geometric properties of a light source

body; (2) description of an optical surface including computation of normals and

the matrix of reflection; (3) parametrization and discretization of the optical surface; (4) discretization of the target surface (the pictorial surface); (5) description of the inverse ray; (6) computation of a ray path after reflection (determination of the inner

ray); (7) testing whether a ray intersects a light source (ray incidence); (8) reading of the ray luminance. At the first stage we specify the following: the equation of luminous

surface of a source in the local coordinate system X's Y's Z's that is F'(r's)=0; the transfer vector -r0 (Fig. 2.3); and the matrix Ms of transition from global to the local coordinate system associated with a light source. The equation of luminous

66

Fig.2.3. Principle of calculation by inverse-ray tracing 1-exit-pupil plane; 2-reflector; 3-light source

surface in the global coordinate system XYZ is known, that is F(S) = 0, hence

.;`1

os

sss

rrSrMr

Discretization of optical surface (the third stage) is a necessary step in creating a numerical model based on the inversc-ray method. Since the grid cells arc chosen, the error of computation depends on the numerical interpolation and integration methods being applied. Note, that rather than dividing a surface into elements vu , we may apply discretization of an exit pupil area ∆Scosθ [see Eqs. (2.35), (2.36)]. This relates to the imposition of a rectangular grid on the exit-pupil plane projection observed from the direction 0a , followed by scanning the grid rows and testing if nodes are luminous or not (Fig. 2.3). An algorithm of searching for luminous points, actually, is similar to one that has been applied in television; the latter is based on the image expansion into an ordered matrix of luminous spots arranged into the rows and columns (Fig. 2.4). Similar algorithms, known as «ray-tracing» algorithms, are widely used in computer graphics for picturing various photometric objects [21-23].

Stages 5-8 are being carried out within every cycle of ray tracing and are destined for calculation of radiance factor ksw and the

67

luminance at given direction. Here follows the algorithm for calculating these values:

Step 1. We find the reflected ray (signs i, j are omitted) at the nodal point of reflector

00 ˆ)ˆ,ˆ(ˆ2ˆ anana , (2.41) or in scalar terms

,2;2;2

0

0

0

zzz

yyy

xxx

aspnaaspnaaspna

where )ˆ,ˆ( 0 nasp . Step 2. We test whether the reflected ray a intersects a light source. Here,

the equations for the reflected ray and a surface have to be solved simultaneously

,ˆlars 0)( sF ,

where a is defined from Eq. (2.41); r is the nodal point vector; l is the ray parameter (see Ch. 1).

To establish incidence of a ray is often sufficient when dealing with sources of uniform luminancc. If ksw=1 and radiation of a source is perplexed, then one more step is needed.

Step 3. To find the point of intersection and calculate the luminance by using the radiance indicatrix ),( ssss LL . In this case, we have to turn into a light-source coordinate system by

Fig. 2.4. Algorithm for searching flashed points over reflector

(φ0≤φ≤φend, 0≤φ≤3600); direct tracing; inverse tracing

68

applying matrix transformations and find spherical coordinates of a ray a (Fig. 2.3), and then get the luminance values from appropriate tables or formulas.

Algorithms for Surfaces of Revolution

We consider an optical system with a specularly reflecting surface having a shape of a surface of revolution (Fig. 2.5). Let OZ be the geometric axis of reflector; dS be the mirror element of the reflector or the point of reflection (the calculated position of dS is in the right half-plane ZOY); xo=0, yo, zo be the coordinates of the element dS centre; φ and a be the polar angles for incident and reflected rays; δ be the polar angle of the normal to the mirror element, due to symmetry of reflector the normal n lies in the plane ZOY.

The angles φ and a are related according to the law of specular reflection

2 , here the angles are positive when counted at clockwise direction. The

viewing vector in the axially symmetric system becomes Taaaa ]coscossinsin[sinˆ0 ;

the normal vector is Tn ]coscossinsin[sinˆ .

From these equations it is obvious that any relationship between 0a and n keeps unchanged provided that [24]:

(1) the point of observation is fixed: 0 , a=const, but the ray of observation moves along the parallel φ=const, ψ=var;

Taaa ]cossin0[ˆ0 , Tn ]coscossinsin[sinˆ .

Fig. 2.5. Parameters of specular reflection surface of revolution

69

(2) the point on reflector is fixed: φ=const, ψ=0, but the point of observation moves in such a way that φ=const, β=var, i.e.,

Taaaa ]coscossinsin[sinˆ0 , Tn ]cossin0[ˆ . Existence of mapping a=a(φ), β=ψ that characterizes ray paths in symmetrical specular system, enables to rearrange the latter relationships in terms of variables ψ and β. This situation is illustrated in Fig. 2.6. The observer leaves the initial point and moves along the parallel a=const viewing the element φ, ψ under various angles β. (The scheme is efficient when a lighting fixture with regularly diffusing reflector is under design).

We note, that if a source shows complex luminance performance, then the only way to find the luminance distribution over elementary mapping (EM) is to scan by using inverse-ray tracing. The most important role here plays a remark made by V.D. Komissarov: an outer-field ray, i. e. a reflected ray, is one-to-one related with an inner-field ray by the law of reflection, hence, if reflected ray is prescribed, the corresponding incident ray is defined as well. First of all, it is advisable to deal with the rays of inner-field. We present Komissarov’s scheme for determining ray luminancc within the EM for high-pressure mercury fluorescent lamp (Fig. 2.7):

Fig. 2.6. Cone of observation [ray A(α,β)→A']

Fig. 2.7. Scheme for determining ray luminance in course of inverse tracing

70

(1) to fix the direction a; (2) while assigning various β, to define the outer inverse ray

aa sin),0( TaA ]cossin[cosˆ ;

(3) to determine the inner inverse ray nnAAAI ˆ)ˆ,ˆ(2ˆ`ˆ ,

(4) to calculate the polar φL and asimuth ψL angles of the inner ray;

yxLxyL IIIII 222cot,/tan ; (5) to determine a point where ray intersects lamp surface, and to find the

luminance by using the indicatrix and obtained values of φL and ψL (Fig. 2.8). This scheme is included in the program based on the inverse-ray principle;

so, there is no need to prepare tables of luminance distribution over the EM trace [25]. The EM boundaries arc determined in β-cycle automatically. A point where the luminance becomes equal to zero indicates the boundary value β bnd.

There are some algorithm modifications based on inverse-ray principle, so the choice has to be done by taking into account different, often conflicting requirements, such as the universality and the most rapid operation.

To organize the programmer’s work effectively, it is advisable to have a set of standardly arranged interchangeable modules that perform similar functions, c. g. geometric modules that calculate the luminous intensity of a tube source. Below, we describe the algorithms for computing the luminous intensity.

Fig. 2.8. Reading ray luminance

from indicatrix of mercury vapor fluorescent lamp

(MVFL)

71

1. Discretely discrete variant The search of flashed area (the source image) is being carried out by

making fixed steps wife respect to φ and ψ (Fig- 2.9). After passing to the spherical coordinates in Eqs. (2.38) and (2.40), we find scalar product )ˆ,ˆ( 0 na :

)cos(sinsincoscos)ˆ,ˆ( 0 aana . For the surface of revolution the polar angle δ and the Jacobian H arу the

functions of argument φ, hence, φ-cycle should be designed as the most rapid and internal; while φ-cycle should be external. By analogy with Eq. (2.40), we obtain estimate )ˆ( 0aI

j

ijswj i

iiijswi

ii LkHaLkHahhaI )cos(sinsincoscos)ˆ( 0

.

Generally, for arbitrarily positioned source the search along the parallel φ=const has to be run from ψo=0 to ψk=360°. In doing so, double contribution of equivalent points ψ=0 and ψ = 360° must be avoided.

Fig. 2.9. On determination of flashed area (image) for symmetrical surface with coaxial cylindric source (a=6°, β=0)

72

Since the surface of revolution is given, it is advisable to prepare in advance the tables or arrays which contain the values of Hicosδi, and further just read and interpolate the needed quantities. Scanning along the parallel φ=const (Fig. 2.9) with the steady step hψ enables to design a calculation program applicable to light sources with arbitrary luminance distribution.

Here we use the concept of filling factor known from the theory of projector design [26].

Assuming a to be small, we obtain from Eq. (2.38) hhkHaaI

jsw

iii coscos)( .

By definition, the filling factor of a zone is /swzc .

For discrete series of ksw values, we obtain

11

Nnhk

c sswz ,

where ns is the number of flashed points along the line φ=const; Nφ is the total number of steps in β-cycle.

Hence, we obtain .coscos)( hcHaaI ziii

2. Discretely continuous model The program will be more accurate if an iteration procedure is applied in

ψ-cycle for searching the image boundaries. For φ =const, we find the derivative of I with respect to φ [Eq. (2.38)]

up

l

LdLdanJIH o

cos)ˆ,ˆ(1 (2.42)

up

l

dLa

)cos(sinsin .

where integration limits are determined in each ψ-cycle. When all ψ-cycles are performed, the total luminous intensity is

73

determined by summing up all the «zonal» candlepowers

N

iii hHJaI

10 )ˆ( , (2.43)

where 1)][( 0 kN ; k and 0 are the boundaries of reflector. When tracing the rays backwards along the line const with the step hψ, the search is being continued until the ray meets the source image. Discovery of the image edge is indicated by transitions

),0()1();1()0(

swsw

swsw

kSkSkSkS

where S(...) is a search state. If we take current value of ksw and its foregoing value kp, then each state of

the search can be coded by a number ,2 pswF kkk

while

.3)1,1(;0)0,0(;2)1,0(;1)1,0(

F

F

F

F

kSkSkSkS

When the first two cases occur, the procedure of computing the roots of equation has to be included into the program. Subroutine ZEROIN from Forsythe’s collection is the most suitable. The mentioned subroutine is based on the bisection method, reliability of which is urgent for the case, when nothing is known about convergence of the process.

In the third and fourth cases the search is being continued until its state has changed or the upper boundary ψ=360° has been reached. A step, with which the search is conducted, is selected empirically; it should not be too small, otherwise the seach will be too time-consuming; and it should not be too large, or else the target can be «missed».

The scheme of computation, when applying Eqs. (2.42) and (2.43), is suitable for any arbitrarily directed light source with prescribed luminance distribution. If a light source has axial symmetry, and

74

Fig. 2.10. Flashed area in case of transversal cylinder (a=0, β=0)

Fig.2.11. Flashed area in case of transversal cylinder (a=60, β=0)

75

Fig. 2.12. Flashed area in case of transversal cylinder (a=60, β=900)

symmetry axis coincides with one of coordinate axes, then the image symmetry should be accounted. Taking a tube which is perpendicular to the symmetry axis, e. g. for a=0, we obtain the image having two axes of symmetry; these are ψ1=0 and ψ2=90° (Fig. 2.10). When α≠0 and the plane of observation is defined by βup=0, the search has to be performed from ψ=0 up ψup=180° (Fig. 2.11); for βup=90°, from ψ =90° up to ψup=270° (Fig. 2.12).

For a source with uniform luminance L=L0=1, Eq. (2.42) can be transformed to the following form

)]([coscos1

)()(m

k

kl

kupi iaJ

)],sin()sin([sinsin1

)()(

m

k

kl

kupia

where the chance that several images m exist along the parallel ψ=const is accounted.

We note, that the total luminous intensity can be calculated by applying efficient procedure of numerical integration [27], which

76

allows (when Jφ does not change sharply) to use rather large step of quantization (see remarks of V.D. Komissarov [24]).

3. Analytical algorithm For some special cases which have theoretical and practical significance, e.

g. for a surface of revolution with a thin coaxial cylinder of uniform luminance, the inverse-ray tracing procedure enables to find analytical expressions for image boundaries at the equatorial plane ψ=0 [28]. The first model of observation is applied in order to determine the boundary ψ(φ) when the ray of observation moves along the parallel φ=const until reflected ray becomes tangent to the surface of cylinder. The condition of tangency of the ray )0ˆ,ˆ( anc to the infinite cylinder of radius rc yeilds a square equation in s=cosψ:

sapspaasP 4sin2sin5,0)2sin(sinsin)( 22222222

0sinsin)2coscos1( 222222 asap , (2.44) where p = rc/r. Flashed points in section ψ=const are defined by inequality P2(s)>0 (these

points lie above the axis in Fig. 2.13a). For a source of infinite length there are two roots of equation (2.44). For a tube with finite length we have two equations that define the upper φup and lower φl boundaries in the meridian plane ψ =0 (Fig. 2.13b)

0)2sin()2sin( aQafup ; (2.45a)

0)2sin()2sin(1 aQaf ; (2.45b)

Fig. 2.13. On determination of flashed points for

α=const in equatorial (a) and meridian (b) planes (- - - — α<0)

77

where

)/arctan( cc lr ; rlrQ cc /22 . Equations (2.45) are transcedental and can be solved by numerical methods

[27, 29]. An arbitrary point φ pertaining to a multitute of flashed points (the image) has to meet the following conditions concurrently

0)( upf ; 0)( lf , (2.46) where and fup, are calculated from Eq. (2.40). To complete the description of boundaries we should have carried out a

detailed analysis for the second image (Fig. 2.13a and 2.14). But the symmetry eases the algorithm. The solution in Eq. (2.44) is mirror-symmetric with respect to angle α, that is

).,(),( )1()2( constaconsta (2.47) For the plane ψ=π, the analysis of image equations being similar to Eq.

(2.45) shows (Fig. 2.14> that )0,(),( )1()2( aa ,

where )1( is the solution of Eq. (2.45).

Fig. 2.14. Boundary angles of flashed area for given viewing

direction

78

Thus, the false roots for the second boundary can be excluded by applying inequalities (2.46) if the sign before α is changed to opposite. The following operator is suitable for this operation

IF(FN(FI).LT.0.AND.(FN(FI) * FW(FI).GT.0)) NROOT = 0, (2.48)

where NROOT denotes the number of roots for the second boundary. Now we can calculate the derivative of the luminous intensity at a fixed angle φ. We take Eq. (2.38) with m=2, β =0, )1(

up = 0, )2(l =π:

)]}(][{coscos )2()1( NROOTaJ i

)]sin(sin[sinsin )2()1( NROOTa , (2.49) where NROOT=1 if the conditions of Eq. (2.46) are satisfied; and

NROOT=0 in the opposite case. We cite the main steps of the algorithm for one a-cycle. Step 1. Using Eq. (2.45), find the boundaries φup(0),φl/(0). Then in the

cycle for φi=φup+(i-1)hφ do steps 2-4. Step 2. Taking the first root from Eq. (2.44), find the boundary ψ(φ). Step 3. After calculating NROOT, go to operator (2.48). If NROOT=1, set

φ(2) =π-ψ(1)(-α), then go to step 4; if NROOT = 0, proceed to the next step. Step 4. Applying Eq. (2.49), find Jφi Step 5. Calculate the integral of luminous intensity [see Eq. (2.43)]

iiii sHaHiaI sinsincoscos , where ci and si are the multipliers that stand before cosδ and sinδ in Eq.

(2.49); the value of Hi can be calculated. Organization of Programs The inverse-ray method enables to reach high flexibility of application

package, where each separate module (subprogram) can be used in different programs. For example, the geometrical module PIPE, that varifies intersection between a ray and cylindric source, can be used in described-above searching algorithm. Since this module is addressed each time the inverse ray is traced, it should be specialized for a certain source type (uniform tube, uniform sphere, metal-halide lamp, mercury vapor lamp, etc.)

79

We consider three programs for calculating luminous intensity, which are based on inverse-ray method: TASDD, TASDN, and TASAN (The Tube: Axially Symmetrical System — Discretely Discrete Variant; Discretely Continuous Variant; Analytical Variant). Specialized program for symmetric optical system shows a better performance than the universal program. This fact is clearly seen from Table 3, where programs TASAN and TASDD are compared.

Thorough logic of TASDN allows to use larger step of quatizalion hψ without loss of accuracy but with gain of time. Furthermore, this leads to small increase (by 10 KByte) of required memory. The program TASDN in comparison with TASDD uses more simple formula (2.38) for uniform sources, but, as it was noted, the likely scheme can be used for nonuniform sources as well.

Table 2.2 Characteristics of programs for calculating luminous intensity produced

by optical system with a tubular light source

Algorithm lime of calculation, rcl. val. Memory volume, KByte

TASDD 3.44 56 TASDN 1.73 68 TASAN 1.00 64

Cylindrical system The scheme of calculation used for surfaces of revolution can be easily

extended onto other surfaces, for instance, onto cylindrical surface. Arithmetic and luminance program modules can be used without changes. Every point of cylindrical reflector is assigned by coordinates φ, x (Fig. 2.15), where the linear variable x plays the role of φ.

If the profile curve is specified, say r=r(φ), then the expressions for n and H take the following form

;cossin0ˆ Tn ),(1cossin rH

80

Fig.2.15. Flashed area of cylinrical reflector

where the angle δ of a normal is defined by equation

ddr

r1)tan( .

Applying Eq. (2.32), we obtain the expression for luminous intensity

up

l

up

l

x

x

dxxLdanaI

.),()ˆ,ˆ()ˆ( 00 (2.50)

Substituting the expression for the scalar product ( an ˆ,ˆ ) into Eq. (2.50), we get

up

l

up

l

x

x

dxxLdHaaI

),(coscos)ˆ( 0

up

l

up

l

x

x

dxxLHa

,),(sincossin (2.51)

where φ1 and φup are the lower and upper boundaries of an image in the transversal plane.

We find now the equations for the flashed area at the transversal plane β=0 when uniform tubular source is used. Assuming that a source image lies within the outline dimensions of reflector, and considering Fig.2.15, we obtain the equations for calculating distances

81

H1, H2 and the difference ∆H:

,);sin(

);sin(;;

12

222

111

202

101

HHHarH

arH

(2.52)

where α01, α02 are the orientations of axial rays at the points φ1 and φ2 . The first two equations in (2.52) prescribe the points φ1 and φ2 implicitly

and can be solved by numerical approximate methods (note, that γ=arcsinR/r). Having found φ1 and φ2, we proceed to determining r1 and r2, and then H1, H2 and ∆H.

The inverse-ray method is the most suitable for calculating the spatial characteristics of luminous field created by specular reflectors [1, 30, 31]

dAФALE p )ˆ()ˆ( ,

where )ˆ(AL is the luminance of reflector along the ray A (Fig. 2.16);

)ˆ(AФ is the value function at the direction A ; Ω is the solid angle associated with reflector flashed area seen from the point P (Fig. 2.16a). Obviously, similar expression can be obtained for Ev, if pN is assumed to be the unit vector being in parallel to OY (Fig. 2.16b). Spatial irradiance can be defined as the irradiance of a plane being perpendicular to the axis of elementary beam, i. e., when pN = A (Fig. 2.16c).

For horizontal irradiance ANAФ p

ˆ,ˆ)ˆ( ,

where pN is the unit normal vector to the horizontal plane (Fig. 2.16b). For automated calculation of Eh produced by cylindrical reflectors we use

the following expression (see Fig. 2.16a)

.)cos(

)ˆ,ˆ(cossin

)ˆ(max max

0 02

dxdrNA

RAA

ALE p

x

x p

yxh (2.53)

82

Fig. 2.16. On determination of spatial characteristics of illuminance produced by specular

reflector

Fig. 2.17. Diagrams of horizontal illuminance produced by parabolic-cylindrical reflector

(a) and luminaire SGS01 (b)

83

The luminance )ˆ(AL is determined by inverse-ray scanning. Figure 2.17 shows the pattern of horizontal illuminance produced by a parabolic-cylindrical reflector and specially designed irradiator, both operating with the metal-halide lamp. The former provides a field of high intensity over small area of 0.8 m2, while the latter spreads the flux over area of 7.8 m2.

2.4. INVERSE-RAY METHOD IN ANALYTICAL REPRESENTATION Determining flashed area boundaries of a symmetric narrow specular zone It would be useful to describe the inverse-ray method by considering

narrow specular zone and its flashed area. The inverse-ray calculating procedure consists of the following stages:

(1) generating inverse ray in the inner space of the optical system; (2) finding the point of intersection between a ray and a light source. We present the law of reflection in the following form

00 ˆ)ˆ,ˆ(ˆ2ˆ anana . (2.54) We observe a motionless mirror element while moving in the outer space

within a cone having the aperture of α. The mobile vector of the ray of observation is

.coscossinsinsinˆ0Taaaa

The normal vector to the mirror element is .]cossin0[ˆ Tn

The matrix of reflection corresponding to Eq. (2.54) takes the following form

2cos2sin02sin2cos0

001

reflM . (2.55)

We define the coordinates of the inverse ray Ta ][ˆ by using matrix product 0ˆˆ aMa refl :

84

;sinsin a aa cos2sincossin2cos ; (2.56)

.cos2coscossin2sin aa The action of matrix (2.55) can be visually represented as imaginary

rotation by 180° of the cone of observation about the normal to mirror element. As a result of this rotation each reflected (in direct tracing) ray matches with the corresponding incident ray (Fig. 2.18). This rotation can be represented as a superposition of two rotations:

(1) rotation about the axis OZ by 180°; (2) rotation about the axis OX by 2δ (Fig. 2.18). These rotations are described by matrices:

;100010001

1

M

.2cos2sin0

2sin2cos0001

2

M

The general rule for composing matrices: each column of a matrix contains the coordinates of the corresponding new basis vector in the old basis. It is easy to prove that

21MMM refl . Let us set up the task of determining the flashed area of a narrow mirror

zone at φ=const. Obviously, the boundary points of this zone correspond to viewing rays that are tangent to the surface

Fig. 2.18. Specular reflection as superposition of two rotations: a - in plane XOY; b - in

plane ZOY.

85

of a light source. We consider some specific cases that have practical significance.

Cylindrical source with uniform luminance Equation for the tangent plane that contains the vector a and passes

through the point A (Fig. 2.19) is cro ry sin ,

or ).( 22222

0 cry (2.57) We transform the expressions (2.56) by introducing the following

substitutions

.2

tan;11sin;

12sin 2

2

2

ttt

tt

After transformations we get

;1

2sin 2tta

;1

sin 2

221

ttkka

,1

sin 2

243

ttkka

where

;sin/)2cos(;sin/)2cos(;sin/)2sin(;sin/)2sin(

4

3

2

1

akakakak

(2.58)

Fig. 2.19. Tangency condition for ray with

respect to cytindrie source

86

Substituting these expressions into (2.57), we obtain 2

212 tkkt , (2.59) where

;tan11 kk ;tan22 kk 22

0

tanc

cr

ryr

.

Equation (2.59) has the roots *

2,1

21112tan

i

ii

kkk

. (2.60)

Where the angles β1 and β2 are counted off from the opposite sides of the axis OY. If 121 kk , then β1=β2=180°; therefore, the narrow mirror zone is completely bright. We find the angle αcr that fits this condition

121 kk , raaa 2tan/sin)2sin()2sin( 2 .

Transforming this equation, we obtain (sec Fig. 2.20) )sin/2)(sin/(sin rrccr . (2.61)

This nontrivial result can be obtained by applying geometrical conceptions of the method of elementary maps (EM).

For α≤αcr, if an arc-tube is long enough, all the elements of narrow specular zone are bright once at a time though they may have nonuniform luminance. We may call it the zone of complete glow. When α>αcr, the image in the mirror zone gels broken no matter how long is the luminous cylinder.

Fig. 2.20. Calculating αcr wilh the aid of spherical triangle.

* It is easy to prove that Eq.(2.60) is equivalent to Eq.(2.44)

87

If we set δ=φ/2 (the zone of paraboloid having its focus at the centre O), the previous formula yields

sinαcr=rc/r. (2.62) Note that this formula is cxact (compare with [26]). Using geometrical terms, we can simplify all considerations accounting the

fact that imaginary rotation of the EM by 180° about the normal of mirror element leads to coincidence of corresponding reflected and incident rays (Fig. 2.21a). After rotation, the planes which are tangent to the EM match as well (Fig. 2.21b). If we pay attention to the rotated axes of the grid, we find out that the tangent planes cut the intercept lan2δ out of the axis OX, and the intercept sin2δ tanyψ out of the axis OY.

Straight-line equation in segments on the plane X'O'Y' being the map of the grid plane is as follows (see Fig. 2.21b)

Fig. 2.21. Principle of correspondence in specular reflection: a - congruent beams AB and

A'B' match after rotation by 180° about normal; b - rotation of cone of observation into internal space; plane XOY transforms into X'O'Y'

(OS is tangent plane to cylinder; OS=sin2δtanψ; O'B= tan2δ).

88

,1tan2sin`

2tan`

YX

or .tan2cos`)2(tan` rXY (2.63)

The cone of observation with aperture α produces a circular trace of radius tdna on the plane X'Y'. Multiplying Eq. (2.63) by cotα, we obtain for the intercepts on the axis Y'

2,1

.tansin

)2sin(tan)2coscot2(sincot`

i

rriY

(2.64)

We denote φ1=2δ-α, φ2=2δ+α. Sines ratios are known* to be the proportionality factors between incident and reflected luminous intensities of a point light source (Fig. 2.22):

Fig. 2.22. Angular dimensions β1 and β2 of cylindrical source image on surface of narrow specular zone: 1 and 2 are point sources being characterized by factors of amplification k1 and k2,

respectively.

* In accordance with the law of enegy conservation, considering pmir= 1.

89

.sinsin,

sinsin 2

21

1 ak

ak

Values 1k and 2k from Eq. (2.58) get the following interpretation: they are the tangent coordinates of the EM trace on the plane of projection (Fig. 2.22).

Luminous filament may be considered as a collection of elements δz. In case of cone-shaped zone, factors ki are common for all elements. Obviously, the zonal luminous intensity subordinates to the law of 1/sinα*.

Here the performance of a specular zone, being a curvilinear surface combined with a thin light source, appears most evidently.

We consider the images of a thin cylindric source in the specular zone of reflector and in the flat specular band having the same width and inclination as a reflector zone (Fig. 2.23). The width of the source image in the flat band is equal to the width 2rc of the source itself, the width of the image produced by curvilinear surface is equal to 2y0 sinβ. For a thin source we obtain approximately (Fig. 2.24)

Fig. 2.24. Possible locations of viewing cone and source image: a - image of ring with gap;

b - image of continuous ring.

* This explains the tendency for sharpening and unsteadiness of luminous intensity curve produced by reflector with linear sources.

Fig. 2.23. Source images within internal surface of narrow specular zone and flat band.

90

that is, the source image lengthens in k1 times in comparison with that of (he flat mirror; reflected luminous intensity increases in k1 times as well. While an arc-tube is thin, the luminance pattern extends almost evenly. Now it is easy to obtain from Eq.(2.64) an exact equation with respect to β. Just note only that the points, where a straight line intersects a circle (α-parallel) of radius tanα, fit the condition

costanx , sintany . (2.65) Excluding x' and y' from Eqs. (2.64) and (2.65), we obtain the quadratic

equation (2.59). It is clear from geometry that for a=acr a secant from Eq. (2.63) turns into

a line being tangent to a circle, and 1k 2k =1 (Fig. 2.24). Image dimensions in meridional plane β =0 or ψ =0 can be determined if

we consider Fig. 2.22. Placing the origin O at the middle of a source with length of 2lc, we find

.0sin)sin(

,0sin)sin(

11

22

rlrl

c

c

(2.66)

Implicit equations (2.66) define image boundaries φ1 and φup, on the positive side of the axis y. Similar pair of equations can be obtained for negative a.

These equations, together with additional equation sinφim=k sinφr, completely define the image of a thin source in angular coordinates φ,ψ.

Sometimes, instead of angular coordinates, it is suitable to operate with Cartesian coordinates on the plane being orthogonal to the line of observation, so to say «pictorial plane». We rotate the coordinate system about the axis OX so, that the axis OZ matches with the line of observation (Fig. 2.25). The matrix of rotation is as follows

Fig. 2.25. Projection of source image onto pictorial plane

91

.cossin0sincos0

001

aaaa

The coordinates x', y' of a point in

pictorial p lane Q (Fig.2.25) are represented in terms of x, y coordinates in the old system:

x`=x ; y`=y cosα-z sinα

The rectangular coordinates of an image for a thin source model are

;2 krx cim

2,1

),sin(

i

iiiim ary (2.67)

where φ1 and φ2 are the lower and upper boundaries that are determined by applying Eq. (2.66). By using Eq.(2.67), the contours of source image and also the lower and upper edges of reflector can be projected on a plane (Fig. 2.26). The luminous intensity of reflected beam for given viewing angle α is defined by the flashed area that is not shaded by a light source.

We estimate an error associated with described thin source model. Wc rewrite Eq. (2.63) in the following form

r

rrk

2

2

sin1sin)sin11(2cossinsin

,

or, denoting sinβ/sinψ = u,

)sin11(2cossin1 222ruku ,

Applying expansion of radicand into series, we obtain

....sin2102cos...sin

2111 222

uku r

We express the variable u in terms of series with small parameter:

rmku 2sin

Fig. 2.26. Exit aperture of reflector with cylindrical source projected

onto pictorial plane 1 - image of lower edge of reflector;

2 - image of upper edge; 3 - image of source;

4 - image of flashed area.

92

and find the correction m:

rrr kkmk

2222 sin...)(22cos...sin

211)sin(

,

whence .2cos2

2

kkm Hence,

;sin)(22cos 2

rkkku

.sin2

2cossinsin 32

rrkkk

Application of formula sinβ = k sinψ leads to relative error

.sin2cos2

1 2r

k (2.68)

Graphs k (δ, α) show limited resources for enhancement of luminous intensity from circular specular zone. The maximum of amplification factor is reached at α=45°+α/2 and is equal to 1/sinα. So, we may consider that with increase of a amplification factor descends according to the law of 1/sinα (Fig.2.27). For δ=α/2 the value of k becomes zero; in this case the reflected beam (in backward direction) runs in parallel to the optical axis without intersecting a light source.

In the first approximation the graph k [δ(φ)] depicts a contour of a flashed area for the chosen angle of observation a (when α>αcr).

Fig. 2.27. Family of amplification factor functions:

1 – α=200; 2 – α=300; 3 – α=400; 4 – α=500; 5 – α=600;

93

Ellipsoidal light source Ellipsoid is specified by equation

(Fig. 2.28)

12

2

2

22

az

byx

. (2.69)

The ray of observation is assigned by the following equations

.;

;0

0

0

zzyy

x

The condition of intersection between a ray and a surface from Eq. (2.69)

yeilds

1)2)(22

220

20

2

2220

20

a

zzb

yy ,

or

.012 2

20

2

20

20

20

2

22

2

22

az

by

by

az

ba

We find the discriminant of quadratic equation

.1} 2

2

2

22

2

20

2

20

2

20

20

abaz

by

az

byD

The expression D=0 is the equation of wrapping cone, where ξ, η, ζ are current coordinates of a cone with the origin at the point (0 yo zo) (Fig. 2.28). Intersection of this cone with the cone of observation rotated by the angle 2δ gives the boundary angles of the flashed zone. We put the wrapping cone equation in the following form

,2

2

2

222

2

20

20

abK

az

by

(2.70)

Fig. 2.28. On deduction of equation for enveloping cone at point of mirror

element A(0, yo, zo); a is viewing vector.

94

where .1// 22

022

02 azbyK

We calculate the right-side value from Eq. (2.70), substituting Eq. (2.58)

2

2243

2

2221

2

22

2

2

2

2

22 )()(4)1(

sina

tkkb

tkktt

aab .

Setting up y0=r sinφ, z0 = r cosφ, we calculate the left side

.cos)(sin1sin

2

2243

2

221

220

20

a

tkkb

tkktar

az

by

Substituting found expressions into Eq. (2.70), we obtain the quadratic equation in u= t2:

,022 CBuAu (2.71) where

,

;5,05,0;

3312231

21

433241243121

4322241

22

fkkfkfkCffkkkkfkkfkkB

kfkfkfkA

and here, in its turn

;/cos 20

20

21 raf ;/sin 2

02

02

2 rbf

;2sin 03 f

2

00

20

22

4 cossin4ra

baf

Since u2=tan2(β/2), the positive solution of Eq. (2.71) is chosen. The points that define the boundaries of flashed area of a narrow zone can be found by solving

2,1)2/tan(

ioi u . (2.72) Let us consider specific cases: (a) Extremely oblate ellipsoid (disk), a=0. We obtain for this case: *

* Note that direct application of the limit α→ 0 in Eq. (2.71) is impossible.

95

Fig. 2.29. Marginal flashed points of specular zone in meridian section for spherical (a) and

disk (b) sources.

,cos21 f ,)/(sin 22

2 rbf ,2sin3 f ;cos4 24 f

(b) Sphere (a = b):

,)/(cos 221 raf ,)/(sin 22

2 raf ,2sin3 f ].)/(1[4 24 raf

We test the obtained formulas by examining the flashed area from axial

direction a=0. In this case, the luminosity of a zone is determined by the sign of discriminant

)].2(sin2cos)/(2sin)/[( 22222 rbraD For a spherical source the luminosity of a zone is determined by the sign of

discriminant (Fig. 2.29a) ).2(sin)/( 22 raD (2.73)

For a disk source (Fig. 50b) ).2(sin2cos 2222 rbD

Taking paraboloid for reflector and matching its focus with the centre of a light source, we obtain, obviously, a completely bright zone, i.e. D>0.

96

Light beam of parabolic projector with spherical sourcc (1) We find the luminous intensity produced by a paraboloid when a light

source is placed in its focus. The equation for a paraboloid with the focus situated in the origin of coordinate system is

).(422 zffyx (2.74) Equation for the normal becomes

Mfyxn T /]422[ˆ ,

where .42 222 fyxM Inverse ray is assigned by the vector

.]cos0[sinˆ0Taaa

The inner ray has a direction defined by the vector ,/]42sin2[ˆ 22 MQMfQyQaxQa aaaa (2.75)

where .cos2sin afaxQa

Obviously, after reflection the axial ray Ta ]100[ˆ0 runs at the direction ax/az=x/z, i. e. passes through the origin of coordinate system (the optical property of parabola). We suppose that the light source has spherical form with a small radius R, where R/f< 1.

If viewing angle is fixed, then applying Eq. (2.73), we obtain the equation for the tangent inner ray

sinα=R/r. (2.76) Keeping in mind the parabola properly r=2f-z, and applying Eq. (2.76), we

find z-coordinate of the boundary bright point of reflector aRfz sin/2 . (2.77)

Substituting Eq. (2.77) into paraboloid equation (2.74), we obtain the equation for cylinder

)1/(4 222 afRfyx . Hence, the luminous intensity at the direction α is

)1/(4)1/(4)( max22 aafLafRfLaI oo

where L0 is the luminance of a light source; αmax=R/f.

97

Since the luminous intensity is nonnegative and αmax is the total angle of radiation, then I(αmax)=0.

Since D>0 in Eq.(2.73), the lighting aperture is seen totally bright in the range of 0≤α≤αmin=R/rmax. Hence, the expression for calculating the luminous intensity of paraboloid with a small spherical light source having the luminance L0 is as follows [26]

,,0;min,1/

;0),2max/(tan

4)(

max

maxmax

min2

2

aaaaaaa

aa

fLaIo

(2.78)

where )2/(cos/ max2

max fr ; max2 is the acceptance angle of reflector. It can be easily proved that at the point a=amin the connection of functions

is provided. (2) We find the luminous intensity of a paraboloid at rather remote, but

still finite distance from reflector [32], Let xo, zo be the coordinates of a fixed point of observation, and

zxzzxx ,;1/0,1/0 the coordinates of paraboloid. The inverse ray is defined as follows

,/]0[ˆ 1000 Mzzxxa T

where .)()( 20

201 yyxxM

The inner reflected ray emerges an angle with the axial ray )/()(tansin 00 zzxxaa . (2.79)

Substituting Eq. (2.79) into Eq. (2.77), we find the coordinates of the boundary bright point of reflector

)/()(2 00 xxzzRfz . (2.80) Expanding the denominator from Eq. (2.80) into series in small values of

x/xo, we get

)1(20x

xaRfz , (2.81)

where 00 / zxa . Substituting Eq. (2.81) into equation of paraboloid, we obtain

.1420

2222

2

0

xaR

afRfy

axRfx (2.82)

98

This is the equation of cylinder shifted from the axis by )/(2 0axfR . The luminous intensity depends on the area of cylinder from Eq. (2.82)

,140

22

2max2

xaR

aaLfI o

where amax= R/f. Equation (2.83) presents an expansion of I(a) into asymptotic series in

powers of l/x0. When x0→∞ it turns into Eq. (2.78), i.e. the expression in parentheses is the null element of series.

The procedure of calculating the luminous intensity at remote distances was comprehensively developed by F. Benford and N. A. Karyakin. Candlepower distribution of paraboloid with a light source placed in the focus is a function of distance r from reflector. The distance of projector light-beam formation l0 is such, that the law of square distances can be applied.

We may assume approximately minmax / axlo , (2.84)

where xmax is the radius of mirror. Equation (2.83) can be presented as follows

202

22 /4 xaRLfII o , (2.85) where additional term defines an error associated with application of the law of square distances. The relative error in terms of the axial luminous intensity is

20

4max

2

2

)2/(tan zaR

. (2.86)

The error is maximal for α=αmin. Combining αmin=R sinφmax/xmax with Eq. (2.84), we obtain

0

max2min

sinl

Ra

Equation (2.86) now transforms into

20max

4

20

)2/(sin4 zl

(2.87)

If we set the permissible error for photometrical measurements δ=δper, then a desirable distance of photometrical testing can be

99

obtained

perper

lz )2/(sin2 max

20 (2.88)

For φmax= 90°, we get a simple formula

perper

lz

0 . (2.88)

Equations (2.87), (2.88), and (2.89) can be used to test the programs of computing illuminance distribution produced by specular reflector.

2.5. CALCULATION OF ILLUMINANCE BY WIENER’S SCHEME According to Wiener’s scheme, the

elementary illuminance at the point of observation A is calculated as a light-vector projection on a small area that contains the said point (Fig. 2.30) [33, 34]

ndLEd )(2 , (2.90) where L is the luminance of a ray hitting a small area; n is the normal to a small area; d is the vector of a solid angle having a

vertex at the point A. Dividing the space on elementary

solid angles and summing up their contributions [Eq. (2.5.1)], we find the resulting illuminance at the viewing point

nLdE (2.91)

We assume the system to possess axial symmetry and direct OX along the symmetry axis. We bring the point A to the origin of the polar coordinate system. Without loss of generality, the following can be set up

Fig. 2.30. Determination of illuminance in accordance with Wiener's scheme

(illuminance at point A is proportional to d cosα).

100

Tnnsn ]sin0[cosˆ ,

ddaaaaad T sin]cossinsinsin[cos , where θn is the polar angle of a normal at the point A; α, β are the spherical

coordinates of a ray. (a) We set θn=0, then the x-th illuminance projection (horizontal

illuminance) is

ddaaLEx 2sin21

, (2.92)

where Ω(α, β) specifies the limits of integration related to the dimensions of a domain percepted as bright from the viewing point A.

(b) We set θn=π/2, then the z-th illuminance projection (vertical illuminance) is

ddaaLEz cossin 2 . (2.93)

Hence, the problem is reduced to finding the limits of integration. It can be solved by using the inverse-ray method.

Calculation of illuminance produced by elliptical reflector Elliptical specular reflector with a light source placed in its focus is widely

used in radiant-heating installations, simulators of thermal flux, and radiative furnaces. For example, the calculation of the thermal field produced by a furnace with a spherical xenon lamp is considered in [31].

If a point source is placed at the focus of ellipsoid, then, according to the ray theory, the caustics point with infinite illuminance occurs at the second focus. In case of a source with finite dimensions, an enlarged source image corresponding to some illuminance distribution appears in the second focus.

Observing the reflector from chosen point, we see flashed only such mirror points that reflect inner rays towards the source. Flashed area corresponds to the solid angle which is cut on a surface of unit sphere surrounding the point on a target plane (Fig. 2.30).

Algorithm of computing the illuminance consists of the following stages: (1) formation of inverse-ray vector

Taaaa ]cossinsinsin[cosˆ0 ;

101

(2) calculation of coordinates of a point where the ray intersects an ellipsoidal surface, and of an inner ray as well;

(3) testing whether the inner ray intersects a light source, and calculating the ray luminance;

(4) calculation of integrals of illuminance from Eqs. (2.92) and (2.93). We consider algorithm of computing horizontal illuminance Ex at the focal

spot of ellipsoid when a spherical source with uniform luminance is used. Stages 2 and 3 can be realized by applying formulas described in Ch. 1. All stages are organized in program cycles: the inner cycle with respect to angle β with a step hβ, the outer, with respect to angle a with a step ha. The choice of step ha, that provides necessary accuracy, is important. Sincc boundary rays arc of the most luminous value, inaccuracy in determining the flashed area boundaries may cause significant errors. Searching algorithm must be organized in two stages, similar to that, used for solution of equations; first, to separate the roots, and then to refine them by applying numerical methods. In this connection, there exists the inner procedure which identifies whether a point is inside (ksw= 1) or outside (ksw=0) the domain.

The boundaries of flashed area can be determined by applying the bisection method until the needed accuracy is reached. We note, that while determining a-boundaries in a-cycles, β-boundaries are not refined, and the search is being continued until the first bright point is found (ksw=1).

Figure 2.31a (curve 1) shows the illuminance distribution at the focal plane of elliptical reflector. The reflector has the following parameters: the interfocal distance 2c=992.4, the focal parameter p=175, the diameter of blind aperture Db=350, the acceptance angle 2φmax = 180°. The diameter of spherical source is 2R=2 (all dimensions are given in milimetcrs). To provide more uniform

Fig. 2.31. Modelling of elliplic reflector with spherical source: a - irradiance distribution in focal spot (1) and in exit aperture of lightguide (2); b - indicatrices for ideal ellipsoid (3) and

system with spherical source (4).

102

illuminance distribution from elliptical reflector, a lightguide is placed in its focus.

Figure 2.31 (curve 2) shows illuminance distribution in transversal section of the lightguide with the radius rs =10 mm at the distance of ls=80 mm.

The indicatrix (candlepower curve) is invariant characteristic of the lightguide. Computation of indicatrix is carried out in the same cycles as in the program for illuminance. The luminance distribution L(β, z0) within the rays cone of a=const is being found in β-cycle for the point z=z0.

The averaging along β gives a value of «filling factor» (as per the second model of observation), or

.),(1)(0 00

dzLzk aa

The elementary luminous intensity produced by the circular ring dz is equal to dI(a)=2π kazo dz0, while the total luminous intensity is determined by integration along lightguide radius:

00 0

0 ),(2)( dzdzLaIrS

a

. (2.94)

Figure 2.31b shows the indicatrices, both for an ideal ellipsoid and a system consisting of elliptical reflector and lightguide.

Calculation of illuminance produced by reflector with plane elements and

Lambertian source In this case, the luminance L being a constant value can be removed

beyond the integral sign in Eqs. (2.91)-(2.93), thus the illuminance calculation is reduced to integration along the contour formed by the rays on the unit sphere

ndLE (2.95) This expression can be put in equivalent form

ndaLE 2/ (2.96) where da is the vector of the planar angle formed by the side-surface

gcneratrixcs of a solid angle cone. For the ease when the contour is specified by a discrete series of rays wc expand the expression

103

for the solid-angle vcclor in lernis of the planar-angle vector. Let two close rays have the polar coordinates (α1 , β1), (α2 , β2). We form

the vectors: ;]coscossinsinsin[ˆ 111111

Taaaa ;]coscossinsinsin[ˆ 222222

Taaaa The vector vp=a1 × a2 has the direction da and the following components:

).sin(sinsin;sincossinsincossin;coscossincoscossin

21213

1212122

2121211

aavpaaaavpaaaavp

(2.97)

The absolute value of vp is equal to sin(da), i. e. )arcsin( vpda (2.98)

where 23

22

21 vpvpvpvp .

Hence, the light vector can be found by summation

i

ii davpvpLE )/(5.0 , (2.99)

where the summation is carried out for each i-th pair of close rays. Since the mirror is flat, the position of source image is exactly determined.

This enables to find angular dimensions of the image when the point of observation is prescribed.

We consider an example of calculating the illuminance from a trough reflector with Lambertian ellipsoidal light source (say, MVFL). The reflector consists of planar plates laid out along the generatrix (Fig. 2.32).

In coordinate system of the source image the latter is defined by equation

12

2

2

22

aY

bZX

In the same coordinate system the position of observer is xo, yo, zo. In accordance with the procedure described in paragraph 2.3, wc find the equation for wrapping cone

2

22222

2

2

20

200

aYbZXK

aYy

bZzXx

(2.100)

104

Fig. 2.32. On illuminance calculation from plane element by Wiener's scheme.

105

where 120

20

20

2 zyxK ; ;/00 bxx ;/00 ayy ./00 bzz

Assuming ;/bXX ;/ aYY ,/bZZ we get the equation in new terms (superposing the cone vertex with the system origin)

00002222 ZzYyXxZYXK (2.101)

We introduce the plane of triangle: an observer – a horizontal element of reflector. The equation of the plane is XmZ . To find the boundary rays in the plane, we consider the intersection between the plane and the wrapping cone. We set up

1rX ; .mZr (2.102) The third coordinate can be determined from the following equation

0)()1( 2000

222 rr YyzmxYmK , (2.103) the solution of which yeilds

.1

)(1(

120

20

200

22

20

20

000

zxxmzmK

zxzmxyY r (2.104)

Equations (2.102)-(2.104) present the coordinates of boundary rays. For special case of distribution along the axis 00 y we obtain the following equations with the preceding notations (without bar)

1

,;)(1

;

20

20

200

2

zxK

mXZxmzmaY

bKX

rr

r

r

(2.105)

The condition of discriminant positiveness in expression for constant K means that the viewing point is outside the ellipsoid. Really, from the condition |x|>b or |z|>a immediately follows that 1)/()/( 22

020 azbx . Obviously, the

range of possible m-values is determined by the condition that the radicand in expression for Y is equal to zero.

Solving the quadratic equation, we obtain

)1/()(20002,1 xKxzm , (2.106)

where K is the same as in Eq. (2.105).

106

In order to find the angular coordinates of the image rays we have to turn into the coordinate system of observer. This can be done by multiplying the vector [Xr Yr Zr]T by the matrix of reflection (see the description of the inverse-ray method), where the latter is defined by the position of the normal to the planar element.

We omit the details, supposing that the reader can restore them easily. We only note that in this specific case ]sin0[cosˆ n , where δ is the inclination angle of the planar mirror to the axis

Now, since xr, yr, zr are the ray coordinates in the observer’s system, the angular coordinates are

./tan

;/tan22

rrrr

rrr

zyxa

xy

(2.107)

To find the illuminance we apply Eq. (2.99) or Eqs. (2.92) and (2.93). When composing a program, one has to take into account that the planar

specular clement plays the role of diaphragm for the observer. Since element bounds form a convex envelope on the unit sphere, it is not difficult to design an algorithm that records if a ray hits within the prescribed domain Ω'. After choosing the direction of going around the integration contour, we register whether a ray occurs within the domain Ω' (Fig. 2.33). The first and the last hittings extract an active contour C2. It is closed by the segment C1 lying inside the source image (Fig. 2.33).

Similar scheme is applicable for calculating the illuminance produced by reflector with other Lambertian sources (e.g. metal-halide lamp). In this case the condition a→∞ must be set in Eqs. (2.100)-(2.104), thus eliminating the compo-nents with Y and y0. The finite dimensions of the source along the Y-axis have to be accounted in final expressions that define the boundary rays. Say, lyr 5.0 , where l is the source length.

Fig. 2.33. Going around flashed area contour of element

107

2.6. THE METHOD OF ELEMENTARY MAPS The melhod of elementary maps (EM) is based upon the following

assumptions [11, 25, 26 ]: (1) a source radiates in accordance with Lambert’s law; (2) the luminance is evenly distributed or slightly varies over source

surface (or its substituted, in the latter case the surface can be divided in a series of small areas with uniform luminance;

(3) the EM trace on a tangential plane is described by a simple geometrical figure (rectangular, ellips, etc.) [35].

The first two assumptions show that a direct ray travel is used within the EM method.

According to the EM method, a point M of reflector is considered to be bright when viewed from direction I if EM associated with this point contains a ray traveling along said direction.

To carry out flashed points extraction it is more handy to operate with a plane. This leads to rather simple equations of EM traces on the plane (assumption 3). The choice of curvilinear coordinate plane, where further operations with EM are being held, is a complicated problem [36].

In Ch. 1 we found out that regular reflection converts any linear combination of vectors into linear combination of their images. Hence, if EM is represented by a discrete set of vectors {S1, S2,…,Sn}, then a trace of EM on the tangential plane is a polygon <A1, A2,…,An>, where

niSMA irefli ,1, . It is easy now to describe a procedure of determining whether an arbitrary

point of target plane is inside or outside the EM trace. The stages of calculating the illuminance by using the EM melhod are as

follows [37]: (1) description of geometrical properties of a light source; (2) description of EM; (3) quantization of EM; (4) quantization of optical surface:

;,1)1( max00 ii Nihi

;20,1)1( ii Nihj

108

(5) quantization of target plane: ;,1,)1( maxmin XxXNkhkx xxk

.,1,)1( maxmin YyYNlhly xyl (6) definition of EM trace on target plane; (7) testing whether grid nodes xk, yi are inside EM trace; if «NOT», then

jump to the next point or node ψi, ψj; if «YES», then record into an array the weight factor lW i

k , which is a projection of solid angle w` on the target plane; (8) calculation of illuminance integral

i i

ikkl lWe .

Description and Quantization of Elementary Map We consider, e. g. the EM of ellipsoid. First, we have to find an equation

of enveloping cone for the light source. The cone vertex is at the point A(x0,yo, zo). A unit vector of the ray AS or AS` may be defined as

Taaap ]cossinsincos[sinˆ , where α, β are the polar and azimuth angles for the mirror point; AO is the

axis. We determine the condition of intersection between a ray and ellipsoid

being defined by the following equation .1//)( 22222 qzRyx

For ease we turn to affine space having presented the transformation in the form given in Table 2.3.

Angular dimension of EM is defined by the formula 2/12

020 )(sin .

Axial ray vector, obviously, is

.0`ˆ20

20

20

20

0

T

p

109

Table 2.3 Cartesian space Affine space

Point x, y, z Angles β, α Ellipsoid

12

2

2

22

qz

Ryx

Ray vector p

Point ;,, Rx

,Ry

,qz

Angles , aRqa tan`tan

Sphere 1222

Vector Taaap `]cos`sinsin`cos[sinˆ

The condition of intersection can be written as follows

sinsin , 2/)ˆ,ˆarccos( `0

`1 pp ,

we have `cos`sinsin[]ˆ`ˆ[sin 000 aapp

,)](`sinsin`cossin 2/120

2000

aa whence

1`)sin`coscos(sinsin)( 200

220

220

20 aa ,

or ).sin1`)(tan1(`)tancos( 22

022

00 aa The equation of enveloping cone is quadratic with respect to tana'

,0`tancos2`tan)]sin1([ 2000

22200 aa

and, returning to real space, we obtain 0tancos2tan)]sin1([ 22

000222

02

02 Raqaq . (2.108) Giving n meanings to β, and solving quadratic equation (2.108), we obtain

a table of values α, βi that define a discrete sequence of EM vectors: T

ii aaas ]cossinsincos[sinˆ .

110

Test of Incidence Between a Point and Elementary Map Trace Significant reduction of testing time can be obtained by defining

rectangular envelope for the EM trace in terms of four numbers [Xmin, Xmax, Ymin, Ymax]. To determine an envelope it is sufficient to look through the list of polygon vertices F=A1, A2, ... , An on the plane:

),,...,,min( 21min nXXXX ),,...,,max( 21max nXXXX etc.

Fig. 2.34. Testing incidence between point W and figure F:

a - point W is outside polygon F; b - point W is inside polygon F. Let R be the ray that originates from the point W and passes though the

polygon F. We may choose the centre of arbitrary segment Ai Ai+1 and draw the ray WAm. Now, we determine the number of intersections between the ray and all the ribs i=1, n-1. If the number of intersections between R and F is odd, then the point is inside the polygon F; if the number is even, then it is outside the polygon F (Fig. 2.34a,b). This algorithm is applied in computer graphics for testing the incidence of a point lo an object, which is approximated by polygon F in pictorial plane. Another efficient algorithm is based upon calculation of angles sum when bypassing a polygon [38].

Modified Scheme of the Method of Elementary Maps

Vogl et al. [37] suggested an algorithm named «the field patch mode». In

this method the aperture is divided into a grid of elemental patches that can be considered as pinholes, and ray beams (EM) correspond to each of these pinholes (Fig. 2.35). The light distribution on the target (pictorial) plane due to energy coming through the pinholes is evaluated, while inclination of corresponding area of mirror surface determines the position of illuminated patch (EM trace) on the target surface. Test grid is selected on the target

111

plane and then conlrubution of elemental patches at nodal points is calculatcd by recording «covering» of points by EM traces. This method enables to consider illuminating beams as being independent from each other. The model is based on the following statement from theoretical photometry [38]: if a system consisting of two arbitrary openings (diaphragms) in nontransparent screens is placed in front of a stretched background of uniform luminance which radiates in accordance with Lambertian law, then the luminous flux ∆Ф that passes through both diaphragms will be equal to the product between the luminance L and the measure of rays set ∆N. It follows from the statement that illuminance at a point of target surface is proportional to the area of illuminating diaphragm [39].

Though computer interpretations of EM method appeared comparatively not too long ago [40], this method was known in lighting practice as far back as 30s.

Pig. 2.35. Scheme of illuminating plane by elementary beams (EM): a - aperture discretization in accordance with VogI et al.;

b - discretization of reflector.

112

2.7. PROBABILISTIC SIMULATION IN DESIGN OF LIGHTING AND OPTICAL FIXTURES

In order to design a reflector with regular and diffused (mixed) reflection,

one can use the methods discussed above, e. g. the inverse-ray method [41]. But when diffusion shows a complex nature [42] or when dealing with interreflections, e. g. in slit lightguides [43], application of these methods is hampered.

A method given below is based on direct mathematical modeling of an object and its operation. Propagation of the radiation from a light source towards a target surface in illuminating installation; or into the outer-space from lighting fixture can be imitated. Obviously, we restrict ourselves to a direct problem, i.e. to calculation of luminous field characteristics (illuminance, candlcpower, etc.), when parameters of installation or lighting fixture are prescribed. This is in contrast to inverse problem, where parameters of optical system have to be calculated to ensure required lighting distribution in a prescribed zone.

Mathematical simulation of a light propagation is based on probabilistic representation of photometric variables. For instance, a luminous flux falling onto a small area A is directly proportional to the probability PA of hitting this area, i. e. ФA=PAФs, where Фs is the luminous flux emitted by a light source. Hence, the problem reduces to finding PA. To solve the latter task, a mathematical experiment has to be carried out: the radiance emitted by a light source is represented by a set N of discrete portions of luminous flux or light rays [44]. Then a random choice is realized from the set, and each ray obtains the initial weight w=1. Next, we trace a traveling path of each ray in a considered space. After reflection from bounding surfaces, the weight of ray w decreases in accordance with reflectance factor ρ. When the i-th ray hits an area ∆A, wi is recorded. After constructing the trajectories for all N rays, we

obtain N the sum

N

iiw

1, the ratio of which to N gives a statistical estimate AP

of desired probability PA, i.e.

N

iiAA wNPP

1)/1( [45].

By applying described procedure, one can obtain the estimates of any photometric variable, c. g. the illuminance estimate averaged over the area ∆A is

N

ii

sA w

ANФE

1, (2.109)

113

and the estimate of luminous intensity averaged within arbitrary solid angle ∆Ω(α, β) (where α, β are the angles that define the axis direction of a solid angle) is

,),(1

N

ii

s wANФI (2.110)

where the sum is determined for the rays leaving a light fixture and appearing within ∆Ω.

Simulation as a process can be divided into three stages [46, 47]: (1) generation of random rays emitted by a light source; (2) calculation of ray paths; (3) registration of rays on target plane. We consider these stages separately, and take for illustration a rectangular

parallelepiped with reflecting walls, inside of which a source with known light distribution is placed (Fig. 2.36). Such object may be regarded as a lighting fixture, where «walls» serve as a reflector, and «floor», as an aperture. Shape of reflecting surfaces has no principal significance.

Fig. 2.36. Scheme of ray tracing within parallelepiped.

114

(I) Generation of rays. For generality of description we use the term of phase-space [48], where every point q is defined in arbitrarily chosen basis

kji ˆ,ˆ,ˆ by the radius-vector r of ray origin, i. e. the point of light emittance from a source or the point of reflection; by the unit vector of direction s ; and by the ray’s weight w. In other words, each point qi, in a space defines a ray, i. e. its location (ri), orientation (si), and power (wi). The choice of qi. i. e. ri, and

is (wi is prescribed or calculated), is determined by the probability distribution P(q) that specifies the location of q within the interval [qo, qk] from the phase-space being considered. In order to calculate q we use the Monte Carlo equation

q

q

qdp0

)~( *, (2.111)

which links the location probability of q within the interval [qo, qk] with the random number uniformly distributed within the interval [0 1]. In this case

)~(qdp is equal to the ratio of ray flux )~(qdФ to the flux Ф carried by all rays from the interval [q0, qk]. Hence,

ФqdФq

q

/)~(0 . (2.112)

Solution of Eq. (2.117) with respect to the upper limit q determines the coor-dinates (r, s ) of a random ray q. Random numbers γ are produced by appropriate generator. Usually, it is a standard computer program [49].

We consider an example of generating a random ray from cylindrical light source with Lambertian side surface (Fig. 2.37). An elementary flux emitted by a small area dzdRdA within a solid angle ddd sin2 is equal to

Fig. 2.37. On modeling of random ray emilted by light source.

* Here and below index marks a variable of integration.

115

dddzdLRФd 2sincos4 , (2.113) hence, the total flux of a light source is

LRlФdФl

lzss

22

0

2/

2/ 0

2/

2/

4 2

. (2.114)

As it is clear from Eq. (2.113), the coordinates of random ray, which carries a flux sФd 4 , are determined by four parameters: the first pair (ψ and z) defines the point r, while the other (φ, θ), the direction of ray take-off from a light source. Since the parameters are independent, after substituting Eqs. (2.113) and (2.114) into Eq. (2.112), the latter splits in four equations. Their solutions with respect to the upper limit of integration express desired parameters in terms of appropriate random numbers γ:

.arcsin;42sin2

);5,0(;2

zlz (2.115)

Further, we express unknown quantities r and s in terms of found parameters

;ˆˆsinˆcos kzjRiRr (2.116)

kjis ˆcosˆ)sin(sinˆ)cos(sinˆ . (2.117) For axially symmetrical lighting fixture, the candlepower of which is given

in the form I(α, β)=I0 cosma, and location of which is specified by vector ro (Fig. 61), the direction of random ray is

kajias ˆcosˆsinˆcossinˆ . (2.118) where

;arccos 1a

ma 2 . (2.119) If, while designing lighting installation, we need to account the joint effect

of all light fixtures, then random choice of a fixture has to be done before generating a random ray. In this case, it is suitable to represent a population of lighting fixtures in the matrix containing m × n elements. Then the choice of a fixture is reduced to drawing a random element, the indices of which i=1, m and j=1, n determine the location coordinates of appropriate fixture, or

116

the vector roij,; (Fig. 2.36). Then, by using a pair of random numbers γm and γn, we calculate these indices according to the following formulas: i=[(m+1)γm], j= [(n+1)γn], where square brackets mean the integer part of a number.

The described procedure can be applied to calculation of luminous intensity distribution of a light fixture as well.

(2) Ray trajectory construction. The sequence of ray states c(q1, q2,…, gi,…) is determined on this stage, i. e. intersection points r and ray directions s after reflections from surrounding surfaces. The appropriate procedure for specular surface is described in Ch. 1.

In case of diffusion the total flux is substituted by a single random ray. In this way the nonterminating branching of rays s can be avoided. Calculation of s here is similar to the generation of a ray emitted by a light source, because reflecting surface may be considered as a secondary source. Generally, the indicatrix of reflection depends on a point location on reflecting surface, as well as on a direction of incident ray s ; this peculiarity must be taken into account. In case of mixed reflection with luminancc factor f ( s , `s ) being symmetrical with respect to the direction of specular reflection Ss, it is advisable [42, 46] to turn to the local basis `,ˆ`,ˆ`,ˆ kji where the unit vector `k coincides with the normal n to the surface at the point of ray incidence, and the unit vector `j is directed along the projection of vector Ss on a reflecting surface (Fig. 2.38). In this basis, the random reflected ray s is defined by the angles of diffusion: the polar angle αp and the asimuth angle βp (Fig. 2.39). Generally, the parameters αp and βp are dependent,

Fig. 2.38. On calculation of reflected ray.

Fig. 2.39. On determination of divergence angles αp and βp.

117

that is why their expressions in terms of random numbers γα and γβ are defined by applying conditional probability [48].

For obtained αp and βp the coordinates of vector s in the local basis take the form

,cossinsincoscos

;sinsin;cossincossincos

00

00

pppz

ppy

pppx

aass

aas

(2.120)

where δ0 is the angle of ray `s incidence (Fig. 2.39). For the indicatrix f ( s , `s ) being symmetrical with respect to n, i.e. when

αp and βp are independent, s is defined from Eq. (2.118). The weight of reflected ray is equal to w=p(r, s )w', where p(r, s ) is the

integral reflectance of a surface at the point of ray incidence r at the direction s , w' is the initial weight.

Important procedure at this stage is the search of a surface, with which a ray intersects (r, s ). For rectangular parallelepiped the condition whether a ray hits a ceiling (C), walls (W1-W4) and target plane (TP) are given in Table 2.4, where rx, ry, rz are the coordinates of a point r where a ray intersects with appropriate plane.

Table 2.4 Conditions of ray’s hitting the plane

C Wl W2 W3 W4 TP 0zs xr0 bry 0

0ys

xr0 crz 0

0xs bry 0

crz 0

0ys

xr0 crz 0

0xs bry 0

crz 0

0zs xr0 bry 0

Obviously, to hit an axially symmetrical reflector bounded by the exit aperture of radius Rcx and the throat of radius Rth a ray must fit the following conditions

,222cxyx Rrr 0zs (2.121)

,222thyx Rrr 0zs (2.122)

The trajectory terminates in the following cases: (1) a ray hits the prescribed surface (the space); (2) the condition w ≤ wmin is fulfilled, where wmin is a minimal weight of a ray set up in advance. When there is a need to take into account a reflection from the target surface, then even after the ray strikes the target surface, path construction is continued.

118

(3) Registration of rays. As a rule, the illuminance distribution E(x, y) on the target plane or the candlepower distribution serves as the main output characteristic. In order to calculate statistical estimates of these characteristics wc divide the target plane or the light fixture in cells ∆A (∆Ω). After tracing all

of N rays, a sum

N

iiw

1, is accumulated in each cell. Further, by applying

Eq.(2.109), or Eq. (2.110), we calculate the jE or jI distribution. Characteristics like efficiency of a light fixture, luminous flux utilization in lighting installation, luminance of reflecting surface, projections of light vector, etc. can be calculated in similar way. To evaluate the accuracy of results obtained by Monte Carlo method is very important. The matter is, that the results, c. g. E or I are not calculated analytically, but they are statistical estimates of E or I , correspondingly. Dispersion DE of desired value, absolute error ε, and N are connected by the following relationship

22 / DEtN , (2.122)

where t is a coefficient that depends on the chosen fiducial probability μ. Dispersion is determined by the following formula

2

11

22 1

11 N

ii

N

ii

s wN

wNA

ФDE , (2.123)

Fig.2.40. Calculated candlepower curves under specular (l), mixed (2) and diffuse (3) reflection

(luminous flux is equal to 1000 lm).

119

where sums iw and 2iw are being accumulated during the process of

calculation. From Eq. (2.123) we see that in order to obtain a sufficient accuracy, a

great number of rays has to be used, and this is time-consuming. Therefore, to decrease DE is very important. Various methods of lowering DE are described in [46].

A. A. Korobko developed a program based on described algorithm. The program enables to compute a candlepower distribution (Fig. 2.40) for prescribed profile of axially symmetrical reflector, which may have three types of reflection (specular, diffused, and mixed), and for the different lamp types (metal-halidc, high-pressure sodium, mercury-vapor).

2.8. CALCULATION OF LIGHT DISTRIBUTION FROM

LAMBERTIAN SOURCES Designing illuminating and irradiating installations, we must be able to

determine such characteristics of light sources as luminous flux, luminous intensity, and illuminance [50]. Description of light field in terms of light-tubcs enables to solve photometric problems easily [51].

Ellipsoidal source We consider the calculation of characteristics of

Lambertian ellipsoid having half-axes α and b (a>b) (Fig. 2.41).

Luminous flux. We find the expression for ellipsoid area and flux. Equation of ellipsoid in parametrical form is

,costbx .sin taz An area element is

.cossin 222222 dttatbdzdxd An area element is

,sin122 22 dtteabdxdS where 2222 /)( abae .

Fig. 2.41. Projection of ellipsoid along direction θ.

120

Further 222 )/(12 zaeb

dzdS

.

A surface area from z = z1 to z=z2 is

a

ezdaez

eabS

z

z

2

1

2/12

2,1 12

)]arcsin()[arcsin(1112 12

211

22 ewew

ewwwwab ,(2.124)

where ./;/ 2211 azwazw The total area is

).arcsin11(2 2 ee

eabS

Hence, the total flux of Lambertian ellipsoid having luminance Lo is

).arcsin11(2 22 ee

eabLФ o (2.125)

Luminous intensity of ellipsoid Inclined projection of ellipsoid on a plane yeilds the large axis (Fig. 2.41)

222 cot` baa . A projection area is

.cossin` 2222 babbaS The luminous intensity at the angle θ is

,2cos BAbLSLI (2.126)

where .2/)(;2/)( 2222 baBbaA

121

Light vector For ellipsoid having a centre at the

point (x, 0, z) (Fig. 2.42) we have the folowing wrapping cone equation

2

22 az

bx

,cot 2

2

2

222

ab (2.127)

where ;1//cot 22222 bzax ,, are direction cosines of a ray

starting from point (x, 0, z) and being tangent to ellipsoid.

Setting 0 , tan/ in Eq. (2.127), we obtain polar angles for boundary tangential rays (Fig. 2.42):

2,1

222222

22

,tan

i

ibazbxaxz

bx

(2.128)

or

2,1

22

.cot

tan

i

i abxzbx

(2.129)

Equation (2.129) can be written in equivalent form

2,1

22

22222

.)(

tan

i

i azxaazbxz

(2.130)

According to the definition, the light-vector is directed along the bisectrix of an angle at the vertex (x, 0, z). We obtain

.)()(

2)tan( 222221 bxazxz

(2.131)

Fig. 2.42. On construction of light-lines produced by ellipsoid of uniform

luminance, and on calculation of absolute value of light-vector at point A;

Hyperbola G divides fluxes into Фup and Фl.

122

Moreover, we have an equation for cofocal hyperbolas

,12

2

2

2

ub

xua

z

where 222 aub . A tangent to hyperbola or polar angle tangent is

,tan 2

2

uaub

xz

dzdx

k

whence

),tan(22tan 21

2

22

2

22

uaubz

ubuax

xzk

since hyperbola equation yeilds ,)/()( 2222 zuaubua .)/()( 2222 xubuaub So, the light-vcctor is tangent to a hyperbola

which defines light-lines of ellipsoid. Focus of hyperbola is 22 baz f .

Here follows an algorithm for finding light-lines in arbitrary point at meridian plane (x, 0, z) and flux fractions radiated at upper and lower hemispheres Фup, Фl (Fig. 2.42, see also [381):

(1) we find hyperbola parameter it by solving the equation 2222222 cot)( 2baubazxu ;

(2) we calculate a point of intersection between a light-line and ellipsoid

;

,

222

20

22

2

2

20

babu

bx

baua

az

(2.132)

(3) setting up ,11 w azw /02 in Eq. (2.124) we find Фl, and Фup= Ф1- Фl,.

To obtain the absolute value of light-vector we have to determine half-axes A and B of elliptic disk in the plane being orthogonal to

123

the angle bisectrix at the vertex (x, 0, z), e. g. by using Fock’s formulas [52]. Generally, light sources, e. g. mercury-vapor fluorescent lamps (MVFL)

possess nonuniform luminance. In this case, calculation of illuminance can be organized by using the inverse-ray algorithm. We put the origin of polar coordinate system into the test point and organize a-and β-cycles (see example in paragraph 2.5).

If inverse-ray scanning is applied, the boundaries a1 and a2 are calculated from the expressions

,bx ,01 a 222 2arctan

azbza

; (2.133)

,bx 21,aa - (see Eq. (2.116)) The value )( fa is being determined on each step inside the a-cycle

by solving the quadratic (with respect to cosβ) equation (2.108). If bx , then the upper limit of integration along β is equal to π. If L(α, β) is the ray luminance, we obtain the light-vector components [see Eqs. (2.92) and (2.93)]:

.cos),(sin

;),(2sin21

0

2

0

2

1

2

1

daLaE

daLaE

aa

ax

aa

az

(2.134)

Filament Light Source We consider a filament with uniform luminance Lo, diameter δ, and length

2l. We find vertical Ez and horizontal Ex components of a light-vector at the point Mo(xo, yo, zo) (Fig. 2.43). We determine the unit vector directed from the source element dz towards the point Mo:

.)(

0)(

ˆ2

020

02

020

0

T

ozzx

zzzzx

xm

(2.135)

The light-vector at the point Mo (Fig. 2.43) is

oo mzzx

dzLdE ˆ)(

sin2

020

, (2.136)

124

Fig. 2.43. On calculation of light-vector

produced by filament source with length of 2l.

where om is calculated in accordance with Eq. (2.101). Further we find

;4)(

2)(22

0222

020

000

000 2]2)(2[ lzlzx

lzxLdzzzxLEl

lz

zozox

(2.137)

.2222

arctan22

12242)222(

222

2]2)(2[

lozox

lox

oxlozlozox

lozox

oxl

oxoL

l

l zozox

dzoxoxoLxE

(2.138)

Light-line produced by Lambertian filament is hyperbola, i. e. a vector having components Ez and Ex is tangential to hyperbola. We find the tangent to hyperbola

125

,

;

;1

0

02

2

22

2

2

2

2

kdzdx

zx

ab

bxdx

azdx

bx

az

whence

,00202

202

02

dxdzzxz

kxza (2.139)

.20

1

0022 x

dxdzzxkab

Light-line equation (omitting index «0») fits the condition

.2arctan

24)(

2

222

222222222

lzxxl

xllzlzxlzx

xzEE

dxdz

x

z

(2.140)

Equation for hyperbola passing through the point (xo, 0, zo) is 222 `)/( XzxzaZ , (2.141)

where z' is calculated from Eq. (2.140) ; a, b, from Eq. (2.139).

Obviously, the ratio between fluxes into the lower and upper hemispheres is Фup/Ф1=(l-a)/(l+a) (Fig. 2.44). The luminous intensity of Lambertian filament at the direction θ with respect to the axis OZ (Fig. 2.44) is

sinsin2 maxIlLI o . (2.142) The total flux is obtained by

integration Fig. 2.44. Light-line produced by

filament source with uniform luminance (hyperbola) (a is line origin).

126

,)2cos1(sin2 max2

max dIdIdФ

.)2cos1( max2

0max1 IdIФ

(2.143)

To find the illuminance produced by filament sources with nonuniform luminance distribution, c. g. from mctal-halidc lamp, the inverse-ray algorithm has to be constructed. In this case, the boundaries of search domain are determined by polar angles

0

01 arctan

xRlz

as

c

; (2.144)

;,2/

;,arctan

0

00

0

2

s

ss

c

Rx

RxRxlz

a

where Rs is the source radius, lc is the source length; xo, z0 are the coordinates of the viewing point.

The search boundaries along β are ,01 ./arcsin 02 xRc (2.145)

Lambertian Thin Tube A source of this kind can be treated as a model for metal-halide or quartz

halogen lamps. Transforming slightly Eq. (2.138), we obtain the horizontal illuminance

xz

xz

lxlzlzxlzxIzEx

1arctan1arctan21

4)()( 222222

222

max (2.146)

Often, e. g. when carrying out Fourier analysis, the coordinate representation of Eq. (2.146) is more suitable than familiar expression for E in angular terms.

For a point below tube centre, i. e. z = 0, we have

xl

lxlxIEx arctan11)0( 22max

(2.147)

We assume that the source length is small, (l/x)→0. We expand

127

the first term in Eq. (2.147) in scries with respect lo small values of x/1 .

....)1(11 4222

lxlx

while the second term we present in the form

...)53

1(1arctan1 42

2

xx

lxl

Hence, rejecting small values, we obtain 2max /)0( xIEx i. e. the law of

reciprocal square distances is accomplished. That means that the linear source can be substituted by the point source, provided that found error is accounted [33].

Let a source be of infinite length, i.e. (x/l)→0. The main term in expansion of the first summand is equal approximately to l/l 2, i. e. it tends to zero.

Assuming I=Imax/l to be a specific luminous intensity, we obtain Ex(0)=Io/x, i.e. the illuminance corresponds lo the law of reciprocal distances.

Ring-Like Filament of Uniform Luminance An observer being at the parallel α=const sees the ring of diameter dr in the

form of ellips with the following half-axes ;5.0 rda .cosaab

Centres of ellips and ring may be considered as coincident. Equation of ellips in parametric form is as follows

;sin ax ,cosby where is an angle of so called eccentric anomaly. An arc segment of

ellips is

dbads 2222 sincos , or

deds 22 sin1 ,

where a

baa22

is the eccentricity of ellips.

128

In our case e=sina, so the length of a ring is

),(4sinsin142/

0

22 aEadaaS

where E(a) is the total elliptic integral of the second genus. Luminous intensity of a ring with specific luminance L0 at the direction a

is )(2 aELdIa or . (2.148)

Since 2/)0( E , Ia can be expressed as

2/

00 2/

)(0

diII aa , (2.149)

where ora LdI 0 , i. e. the luminous intensity of a ring is equal to the

average value of indicatrix 22 sinsin1)( ai being calculated along the circular ring (0≤λ≤2π) and multiplied by the luminous intensity at the normal direction.

The total luminous flux is

2

0

2

0

sin)(4sin2 daaaELdadaIФ ora . (2.150)

To calculate Ia and Ф we may use the expansion of )sin( aeE into series [53]

...5642

531342

3112

12/)( 62422

eeeeE

(2.151)

A program for calculating E(e) with required accuracy can be composed easily. The series converges like a geometrical sequence, so determination of relevant number of elements in series can be carried out automatically.

Application to Problems of Reflector Design Geometrical representation of a field produced by a light source leads to

rather simple solutions of photomctrical problems. Direct problem. Let a source be in the form of uniformly radiating thin

tube or filament located along the vertical axis. We have to

129

estimate the flux fraction q which falls on reflector throat if initial point of reflector has coordinates xo, zo, and filament length is 2l.

Here follows an algorithm of calculation; (1) knowing the initial point and applying Eq. (2.140), we find the tangent

to hyperbolic light-line; (2) using Eq. (2.139), we find the half-axis of hyperbola; (3) we calculate q=(1-(α/l)) x 100%. For l=12; xo=100; zo=80, we obtain q=28.1%. Inverse problem. For the same light source and prescribed throat radius xo

and corresponding flux loss, the location of initial point zo is required. When x=xo is prescribed, the derivative from Eq. (2.106) becomes a

function of z. Then the expression z2- x z z' - a2 is a function of z as well. Thus, a desired value of z is the rool of this function. Geometrically, the solution of this problem means finding the intersection point between a hyperbola originated from the point (0, α) and a vertical x=xo.

So, the following algorithm reaches the goal: (1) to find the half-axis of hyperbola a=(1-0.01 q) l; (2) to solve the equation z2 - x z z'

x-a2=0 by using some numerical method (e. g. by using the collection ZEROIN from Annex 1). Allowed range of z has to be set in the program. In order to decrease losses obtained in previous example, we choose q=8% and obtain zo=132.4.

2.9. MODELS OF REAL SOURCES AND MATERIALS Models of metal-halide lamps According to [54], the luminancc distribution along arc-tube of vertical

lamp is described by the following formula 6.02

0 )](sin)/(exp[)1/()( bg rrnhmhLL (2.152) where φ is the angle between the lamp axis and the viewing direction; r is the distance between viewing direction a and the axis of cylindrical lamp body; h is a current height, and h=0 indicates the centre of an arc-tube; Lo, rb, hg are the constants depending on lamp power; Lo is the luminance at the centre of an arc-tube axis; rb is the inner radius of an arc-tube; hg is the apparent arc-tube length; m, n are the coefficients in the relationship between the luminance and the point position at an arc-tube (Tabic 2.5).

130

Tabic 2.5 Data on metal-halide lamps which are necessary for calculation of lighting

fixture

Characteristic Power, W 250 400 700 1000 2000

Lamp dimensions, mm diameter a 91 122 152 176 100 length b 227 290 370 390 430

height of light centre c 142 185 240 245 255 Constant m 0.45 0.21 0.63 0.78 1.15 Constant n 4 3.8 2.7 3.2 3.3

135 2 10.2 14.2 Luminous flux, lm 103 19 35 60 90 190

Luminance at arc-tube centre, cd/m2 x 106 6.3 6.6 7.3 7.4 8.8

For horizontally positioned lamps the luminance along the axis is

uniformly distributed, but the discharge cord shifts upwards from the axis by r0. The luminance model for this case is as follows

5.0200 )](sin/)(exp[)( brrraLL (2.153)

Constants are given in Table 2.6. Table 2.6

Lamp type Parameter value Lo, Mcd/m2 ro, mm α

DRI 250-5 7,1 0,5 4,26 DRI 400-5 6,8 1 4,13 DRI 700-5 6,8 1 2,07 DRI 1000-5 7,0 0,5 1,72 DRI 2000-5 7,6 1 1,25 DRI 3500-5 10,5 1,5 1,20 The presence of sinφ is explained by partial transparency of discharge for

nonresonancc lines of scandium and for visible lines of mercury.

131

Module for luminance determination We describe now the algorithm that

uses Eqs. (2.152) and (2.153) and determines the ray luminance for melal-halide lamp. Applying these formulas, we suppose that the ray luminance docs not depend on the angle of emission, i. e. radiation follows Lambert's law.

Thus the luminance of inverse ray a becomes a function of the point coordinates zL and rL where a ray intersects the cylindrical arc-tube (Fig. 2.45).

Equation for inverse ray is lass ˆ0 ,

where Taaaa ][ˆ 321 , and T

ooo zyxs ][0 is the radius-vector of reflector point.

The condition of tangency between a ray and a cylinder of radius rL yeilds

),()( 22

21

220201 aarxaya L

whence,

.22

21

0201

aa

xayarL

(2.154)

The ray luminance is determined by the following procedure: Step 1. We calculate xo, yo, zo and a1, a2, a3 in the local coordinate system

associated with a light source. Step 2. We calculate the coordinates of a point where the ray intersects the

cylinder. They relate to the roots l1 and l2 of the square equation: l2 : l1 → X, Y Step 3. We find the scalar product

YaXascal 21 , if scal<0 then the intersection point corresponds to the root l1 (see Fig.

2.45) and

Fig. 2.45. On determination of ray luminance for cylindric source

132

.130 lazzL In the opposite case,

.230 lazzL Step 4. By using Eq. (2.154), we calculate the minimal radial distance

between the ray and the axis of an arc-tube. Step 5. We calculate the ray luminance, by substituting r = rL and z = zL

into Eqs. (2.152) and (2.153). We take the values of r, Lo, m, and n from Tables 2.5 and 2.6.

Model of mercury-vapor fluorescent lamp (MVFL) A luminous body in MVFL can be regarded as an ellipsoid of revolution

with major half-axis a and minor half-axis b (Fig. 2.46a). Surface luminance indicatrix of MVFL has one plane of symmetry which coincidcs with the corresponding meridian plane of a bulb [24]. The real indicatrix is not a surface of revolution but still it is substituted by such a surface. The luminance at the direction a is as follows

Fig. 2.46. Model of mercury-vapor fluorescent lamp; a - determination of eccentric anomaly angle; b - reading luminance from bulb meridian.

133

m

md LLL cos (2.155) where Ld is the diffuse component of luminance indicatrix; Lm is the mixed component of indicatrix; γ is the angle between a and the normal to the lamp surface.

Generally, the power of cosine is not a constant value, it varies from 2 to 4, according to [24]. The values of Ld and Lm are the functions of point location on a bulb meridian, i. e. Ld= Ld/(λ), Lm=Lm(λ), where λ is an angle of eccentric anomaly related to ellips coordinates by equations x=b cos λ, z=acos λ . The λ-values are usually calculated and put on the bulb profile similar to marking of a circular limb (Fig. 2.46b).

Calculating light source flux for piecewise linear specification of candlepower curve

Let the function Iφ=I(φ) be changeable between the nodes φo and φ1 linearly

)( 001

1

oo

IIII (2.156)

where )( oo II , )( 11 II . We consider the following cases:

(1) Cylindrical system

We substitute Eq. (2.156) into the flux formula

1

0

dIF

and get ).)}((5.0{ 11 ooo IIIF (2.157)

Formula (2.157) is suitable for programming. It gives a good approximation, and the result needs only small correction.

(2) Rotation ally symmetric system For this case we have

0

sin

dIF .

134

After substituting φo, we obtain

)sin(sin)coscos( 11

0111 o

ooo

IIIIF

or

o

ooo IIIF

1

11011

sinsincos)()cos(cos (2.158)

Calculating mean radius of dispersion indicatrix There is often a need to obtain suitable approximation of dispersion

function expressed in terms of direct component of reflected or transmitted light. The following function is applicable in this case

))2/(sinexp( 2 aii o (2.159) We find the luminous flux under the indicatrix

.))2/(sinexp(sin20

2

daiF o

Translating the sine according to double angle formula and changing the variables, we get

)))2/(sinexp(1)(/4( 2 aaiF o (2.160) Example. Let an angle of dispersion be prescribed, say, 2φ. The task is to

ensure that the flux within the cone of γ=φ is equal to 70% of the flux within the cone for γ=π/2. This states a condition

)).2/exp(1(7.0))2/(sinexp(1 2 aa (2.161) hence, we obtain the equation with respect to x=exp(-a/2)

)1(7.01)2/(sin2 2

xx

. or

,03.07.0 qxx where .cos1)2/(sin2 2 q Assuming the x-value to be very small, we find qx=0.3, whence a=323.06.

135

2.10. GRAPHICAL REPRESENTATION OF LUMINOUS FIELD PRODUCED BY LIGHTING FIXTURE

Computer graphical representation of output characteristics is of great

significance [55, 56], since it enables to preestimate the quality of designed lighting fixtures.

It is a known fact that the light field of a luminaire can be presented by a family of light-lines or lubes. The main property of a light-tube is such, that a constant luminous flux flows through its arbitrary cross-section. The picture of luminous field allows, e. g. to estimate illuminance on any arbitrarily oriented surface.

For axially symmetrical specular surface with coaxial source the luminous field is separated into tubes, the walls of which arc the surfaces of revolution with common axis. To make a picture of luminous field [57], one must calculate a family of illuminance curves E(y) at different cross-sections z=const (Fig. 2.47). Assuming that one and the same luminous flux ∆Ф flows within a space between two lines, we can determine the coordinates of a tube from the following equation

ySФyE /)( (2.162) where Sy is a cross-section area of

a tube. Let the total number of tubes be

N, then the k-th tube coordinates in zi section can be found from equality

kФyФ )( , or

,),(20 yk

оi ФNkdyyyzE

(2.163) where )(yФ denotes the flux determined from the curve of «luminous flux growth» for section zi;

kФ is the summarized luminous flux from k tubes; oФ is the total luminous flux determined by integrating from 0 to 90° (Fig. 2.47).

Fig. 2.47. Determining light-tube coordinates in section zi.

136

Since finite dimensions of flashed area should be considered at close distances from luminairc, the calculation of E(z,y)-curves becomes a laborious task. Inverse-ray principle enables to consider nonuniform luminance distribution of a light source, that is especially important when dealing with high-pressure gas-discharge lamps.

It is a known fact that a specular reflector with metal-halide or high-pressure sodium lamps shows harsh illuminance irregularities in a proximate zone at points close to axis. Generally, integration of such harshly changing functions, while determining tube coordinates [Eq. (2.163)], is a nontrivial problem. Adaptive quadrature program, e. g. QUANC8 [29] solves the problem best of all. The program automatically selects the step of integration related to prescribed accuracy: fine at the vicinity of peaks, and rude at a distance.

Solution of Eq. (2.163) can be found by applying subprograms of inverse spline-interpolation SPLINE and SEVAL (Annex 2).

Below, we give a logic scheme of determining light-tube coordinates in prescribed cross-section; when the illuminance distrubution in the cross-section is found beforehand, e. g. by applying program CAMIL. The scheme can be translated into computer program.

(1) To input: M, N, HY, YMAX, where M is the number of tubes, N is the number of illuminance levels E(y); HY is the step with respect to y; YMAX is the null of function E(y).

(2) To input: E(I), I=1, N, where E(I) is the array of illuminance values. (3) To calculate: E(I)=E(I)*YI, YI=(I-1)*HY, I=1, N, i. e. the value of

integrand function that is recorded in storage location E(I). (4) To calculate F(I), I=1, N. In order lo calculate the luminous flux-

growth function one should apply subprogram QUANC8 [29]. While calling QUANC8, one should set up the lower limit of integration A=0, and the upper limit B=(I-1)*HY, 1=1,N. To compute intermediate values of integrand function E(I), address to subprograms SEVAL and SPLINE.

(5) To calculate the step DF=F(N)/M. (6) To calculate the tube coordinates S(K), K=1,M by using subprograms

SEVAL and SPLINE. To write ARG(I)=F(I), I=1,N into the arguments array; to write FUN(I)=(I-1)*HY, I=1,N into the array of function values.

(7) To print: S(K), K=1,M; F(I)=1,M. Figure 60 shows drafts of luminous field near aperture of specular reflector

with lamps DRI 400-5 and reflector lamps with arc-tubes. The drafts were produced by computer.

From above-stated we may conclude the following:

137

(1) In the proximate zone the luminous field of lighting fixture with metal-halide lamp is characterized by increased solid density of energy. Luminous flux transported through the tube which has a diameter approximately half of that of luminairc aperture makes up to 60% of the total flux;

(2) The z-coordinate of flux concentration zone enables to identify the type of candlepower curve;

(3) The luminous field picture enables to forecast the distance of photometrical testing of lighting fixture. At the test distance light-lines differ but negligibly from asymptotes of a point source;

(4) The least thermal load results for flat safely glass, since the density of light-lines (Fig. 2.48) increases greatly with the growth of glass curvature, and with approaching to the axis.

Fig. 2.48. Luminous field of specular luminaire with metal-halide lamp DRI-400: a - candlepower curve of G type; b - candlepower curve of K type.

138

2.11 CALCULATING LIGHT SOURCE INDICATRIX FOR SPECIFIED LUMINOUS INTENSITY

When geometry and luminance characteristics of a light source are set,

calculation of luminous distribution relates to direct problems. Sometimes an inverse problem arises, that is to determine local characteristics, e. g. luminous intensity indicatrix or luminance indicatrix when integral output parameters, say, candlepower curves, are known. To solve these problems, one has to deal with integral equations [58 ].

Ring-like source Complex spiral luminous body is often substituted by torus or anchor. We

consider a ring segment having the direction ]0cos[sin .

A ray of observation is ]cos0[sin a .

An angle between the element normal and the ray of observation is sinsinsin a . (2.164)

We introduce the condition that specific luminance (per unit of length) depends on the angle γ

L = Lγ. The luminous intensity at the direction a is

,cos dsIdIa or, introducing ring diameter dr,

.sinsin15.0 22 daLddI ra The total luminous intensity of a light source is

.sinsin122/

0

22

daLdI ra (2.165)

Setting Lγ=Lo, we obtain Eq. (2.144). Wc introduce now the indicatrix of luminous intensity

139

,cos

oLL

i

hence

2/

0

2/

00 2/

5.04

di

LddiLdI orra ,

where Lo is the mean luminance at the direction a=0. Then the luminous intensity of a thin ring is

2/

0

2/

00 2/

5.04

di

LddiLdI orra , (2.166)

or

'2/0

2/

0

di

II aa , (2.167)

i. e. the luminous intensity of a ring is equal to the indicatrix value, averaged along the circlc (0≤ψ≤π), and multiplied by the candlcpowcr at null-direction.

Expression (2.167) is an integral equation in iγ. This is Volterra’s equation of the first genus, mainly the Abel’s equation [59] if reduced to canonical form.

We substitute the variable in the integrand in Eq. (2.167) daadd cossinsinsinsin , (2.168)

whence

22222 sinsinsin

sinsinsinsin

2sin1sinsin

cossinsin

ad

aad

ad

ad

Now we recount the limits of integration

asinsin2/0sin0

and obtain

aJadi

2sinsinsin2/

022

, (2.169)

140

where 0/ aaa IIJ . We translate Eq. (2.169) lo the following form

.sinsin

)(sinsin

2/

022

2

aJa

di

(2.170)

Equation (2.170) is similar to that of Abel. Within the function class Ja with continuous derivatives in the interval [0, π/2] equation (2.170) has unique solution [60]:

,sinsin

)(sin`

sin)0(1

sin

2

2cos

022

2sin

2

a

adJJi a (2.171)

whence,

sin

022

2sin

sinsin)(sin`sin)0(a

adJJi a (2.172)

Taking into account that addJaddJ cos/)/()(sin/ , and having made the change of variable wa sinsinsin , after manipulations we obtain

.cos`sin)0(

2/

0

a

dwJJi a (2.173)

If the curve Ja was obtained by measuring, errors are inevitable, and the equation (2.173) becomes unstable due to differentiation. Therefore, we have to use more stable quadratic forms [61].

Cylindrical source We consider a cylinder with radius rc being small in comparison with its

length. Let Lγ be the luminancc of unit area of cylindric surface at the direction γ. The normal to cylinder element is

]0sin[cos . The ray of observation is

]cos0[sin aa . The luminous intensity at the direction a is

141

,cos2/

02/

dLII aa (2.174)

where cosγ=sinα cosψ; Iα=π/2 is the luminous intensity of cylinder at the direction α=π/2 for uniform unit luminance. Reducing the prescribed luminous intensity curve to this value, we translate Eq. (2.174) in the following form

,cos2/

0aJdL

(2.175)

The expression (2.175) is Abel’s integral equation. By changing variables it can be reduccd to canonical form just in the same way as it was done above. The solution is as follows

2/

0 cos`

cos)0(2

adwJJL a , (2.176)

where .sincossin wa Setting aJa sin , from Eq. (2.176) we find 1L . 1aJ , we get Lγ

cosγ = 1. These correspond to variants of totally opaque and absolutely transparent discharges in Gershun’s model [62].

142

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M.: GIFML, 1958.

145

CHAPTER THREE INVERSE PROBLEM IN OPTICAL SYSTEM DESIGN 3.1. POINT-SOURCE METHODS AND ALGORITHMS General Equations The inverse problem means determination of active surface shape of

reflector when intensity distribution or efficiency criterion is prescribed. The inverse problem is considered to be well posed if its solution: (1) exists; (2) is unique; (3) is stable when input data are slightly changed.

Existence and uniqueness of solution for specular surface with a point source were proved by N. G. Boldyrev and V. D. Komissarov [1, 2], Equation for nonsymmetrical specular surface is presented in the following form

,)cosh()cos()sinh()sin()(ln

tadttadrd

(3.1)

where r is the radius-vector of a surface; ))2/ln(cot(t ; ))2/ln(cot(aa .

V. D. Komissarov showed that by changing the variables the equation can be reduced to elliptical equation of Monge-Ampere. Recently, similar result was once again discovered by Schruben [3, 4],

Still, Eq. (3.1) has to be supplemented with a flux equation stating that reflected luminous flux is equal to incident flux multiplied by reflectance factor

ddIddaaI sin),(sin),`( , (3.2) or

sin),(sin),`( IDaI , (3.3)

146

where

.

a

a

D

The sign in Eq. (3.2) can be chosen arbitrarily. Keeping in mind that luminous flux is a positive value, we must take the absolute value of determinant in Eq. (3.3).

Remote Zone Axially Symmetrical Surface In this case we have

.1,0,0

a

Equation (3.1) yeilds

,2

tanln,0ln arr

(3.4)

where φ and α are positive when counted in clockwise direction (Fig. 3.1). Hence, Eq. (3.3) becomes

dpIadaaI sin)(sin)`( . (3.5) This equation assumes existence of two types of surfaces: (1) a surface of elliptical type, where reflected ray intersects the axis of

symmetry (Fig. 3.1a); (2) a surface of hyperbolical type, where reflected ray docs not intersect

the axis of symmetry (Fig. 3.1b).

Fig. 3.1. Elliptic (a) and hyperbolic (b) surfaces.

147

Surface type can be chosen in advance, if bounding conditions (e. g. constraints on reflector diameter) are taken into consideration. Equations (3.4) and (3.5) can be represented in the form

;sinsin

)`()(

aaIIa

(3.6a)

2tan arr

. (3.6b)

The obtained system of equations is a consequence of supposition stating that the laws of geometrical optics arc valid. Similar equations are studied in optics, acoustics, theory of antennas [5, 6, 7]. These are differential equations of the first order. Under initial conditions φ =φ0, a= a0, r=ro the system (3.6) constitutes Cauchy’s problem. Since Eq.(3.6a) has a singularity at a=0, a curve must start from angle a≠ 0.

Expression (3.6a) is an equation with separable variables; it can be easily integrated; thus the function of axial ray paths ao= ao(φ) can be obtained. After substituting into Eq. (3.6b), we gel the quadrature

.2

)(tanexp0

darr o (3.7)

While composing an algorithm for solving the problem stated by system (3.6) or for determination of r(φ) by applying Eq. (3.7), the following stages must be undertaken:

(1) normalization of fluxes, i. e. instead of given I(α), the following has to be considered

),()`( aIKaI am where

daaaI

dIK

k

k

a

a

am

0

0

sin)`(

sin)(

;

here φ is an acceptance angle of reflector, α is corresponding polar angle of reflected ray [the end of I(α)-curve];

(2) the own light of a source has to be taken into account, i. e. )()`()``( 0 aIaIaI is the required luminous intensity curve;

148

(3) function )``(aI is now substituted into Eq. (3.6a).

Cylindrical Surface

For this case we have equations [1, 8] similar to Boldyrev’s equation (3.1)

;)`()(

aIIa

(3.6a)

2tan arr

. (3.6b)

where I(φ) is the luminous intensity of a source, I'(α) is a prescribed luminous intensity in transversal plane.

Equations (3.8), as well as Eqs. (3.6), can be solved by methods of numerical integration or be reduced to approximate quadratures. The source image must rest within mirror length dimension [9], this is a condition of applying Eqs. (3.8). Candlepower curve of a designed lighting fixture is usually prescribed in the form of indicatrix with maximal radius equal to unit:

)(afI oa . Therefore, the luminous intensity in absolute units of

measurement (cd) is )()`( 0 IIKaI aam , where amK is a scale factor. If candlepower of a lamp in transversal plane is oII )( , then the specified curve will be

)1()`( 0 aamo IKIaI , (3.9)

where amK is a factor of maximal amplification. Integrating (3.8a) with account of Eq. (3.9) within maximal acceptance

angle max and maximal divergence of reflected rays maxa , we obtain

maxmaxmax /)( ФaaKam , (3.10) whence

)(1maxmaxmax aФaKam

,

where the function max

0

0max

a

a daIФa can be found by numerical

integration. Thus, assigning φmax, amax and reflectance po, we find the constant Kam

being the term of equation (3.8). Now Eqs. (3.8) are well integrable if a condition of «physical feasibility» is met: .1aamIK

149

Proximate zone Axially symmetric system Similar to flux equation expressed in terms of luminous intensity, we can

write an equation that defines the local balance of fluxes [10, 11] .sin)(2)(2 dIdxxxE (3.11)

Adding Boldyrev’s equation to Eq. (3.11), we obtain a system of differential equations

sin)()(

lxxE

dxd

; (3.12a)

2tan ar

dxd

ddr

, (3.12b)

where relationship between a, x, φo and z is )cos/()sin(tan HRxra . (3.13)

Under initial conditions φ=φ0, x=xo, z=zo Eqs. (3.12) and (3.13) yield desired profile curvc of axially symmetrical reflector presented in the table form, x(φ) and z(φ). The discussion on peculiarities in solving equations like (3.12) can be found in Schruben’s work [4].

00 dEE . (3.14) Cylindrical System For this case the equations defining the profile curve are perfectly similar

to those applied to axially symmetric system and have the following form

)()(

l

xEdxd

; (3.15a)

2tan a

dxdr

dxdr

. (3.15b)

For a light source we take the uniformly radiating tube (or filament) for which

,oII .)1( 12 xEE od

150

where ,/ Hxx ./00 HIE In accordance with Fig. 3.2, where φd is an angle of divergence, we obtain

the following equation for the balance of integral fluxes

dIdxxExEKex

xam

00

)(])1()([ 121

0)0( ,

whence, taking 1)( 00 ExE , we obtain for moc

,01 xx

K dmam

(3.16)

where ,/00 Hxx ./11 Hxx Note, that Eq. (3.16) is similar to Eq. (3.10) by structure. Thus, we have

the following system of equations

])1([ 12

xK

dxd am ; (3.17)

,2

tan axd

drxd

dr

where Kam is calculated front Eq. (3.16). The sign «+» fits the condition of ellipticity (see Fig. 3.2). The condition of

«physical feasibility» is equivalent to condition of positiveness of denominator in the right side of Eq. (3.17): Kam>1, or

dm

xxh

01 (3.18)

(compare with corresponding condition for remote zone).

Fig.3.2. On energy balance in cylindrical system.

151

Singular and Critical Points in Equations of Specular Surfaces Here we consider the conditions of correctness for Cauchy’s problem

solution in more detail, when dealing with specular surfacc equations, say Eq. (3.12). We write the first relationship from Eq. (3.12) as ),(/ xFdxd . For existence and uniqueness of solution the derivative /),(xdF must be bounded within the whole range of φ-values from segment [φ', φ" ]. Therefore, the luminous intensity curve of a light source must be a smooth function with respect to an argument and without naughts within given interval. Almost all real functions meet this requirement. But still, it is obvious that the point φ=0 is a singular point. If it is included in the range of reflector angles, the uniqueness of solution can not be guaranteed.

To eliminate the singularity at null, one can solve the first equation individually, taking into account that this is an equation with separable variables.

Integrating, we obtain the flux balance equation (accounting the direct flux)

0 0

,)}()({)(sinx

xd dxxExEdi (3.19)

where Ed(x) is the direct illuminance at setting range [x0, x1] of illuminance curve.

The flux balance equation defines the relationship implicitly.* By the implicit-function theorem this equation has a unique solution within the whole x-range except for the point where the right-side integrand becomes equal to zero, i. e. ).()( xEdxE

Supposing )()( xeKxE am , we obtain an equation for singular points )()( xExeK dam , (3.20)

where amK is the scale multiplier, e(x) is the prescribed illuminance distribution.

Points that fit the condition (3.20) are the critical points of solution. The scale multiplier can be found from expression

* It can tic solved numerically, e.g. by applying ZEROIN complex (see Annex).

152

1

0

00

)(

sin)()(sin1

`x

x

am

dxxxe

dIdiK

, (3.21)

where φ0, φ1 are the boundary angles of reflector; `1

`0 , are the boundary

angles of direct illumination, i. e. ),/arctan( 0

`0 Hx )./arctan( 1

`1 Hx

Critical points of cylindrical system [Eq. (3.8)] are defined as follows ,1)(0 aIKam (3.22)

where Kam is calculated from Eq. (3.21) [for «isotropic» light source see Eq. (3.10) ].

Duality of the Problem There is an alternative approach to solution of the problem of profile

determination. One may first set an acceptance angle of reflector φmax (or φ1) and then find Kam from Eq. (3.21), where Kam is a parameter in the flux balance equation. Solving Eq. (3.19), we find the ray-tracing function x=x(φ). Hence, the right side of Boldyrev’s equation is defined as a function of variables r and φ [accounting Eq. (3.12) ]. On the other hand, we may prescribe Kam and then find the angle φmax from the flux balance equation or solve the system of equations (3.12).

It can be demonstrated that under the same conditions both approaches lead to the same result.

Presence of critical points change the matter. Exclusion of these points leads to narrowing the range [xo, x1] which specifies the illuminance curve, and, consequently, changcs the parameter Kam. So, according to Eq. (3.21) the angle of acceptance changes as well. Applying iteration method, one can find the Kam value that fits admissible limits.

When designing the shape of reflector in luminaire for proximate illumination, actually, Eqs. (3.15) or (3.18) are used as initial expressions. Further, the relationship x(φ) is determined explicitly by integrating numerically the equation of specular surface. We note that each method of integrating differential equations uses polynomial representations of function within integration interval. For instance, the fourth order of Taylor’s truncated formula is used in

153

Runge-Kutta’s method. Precisely from this point of view the efficiency of integrating methods has to be estimated.

Numerical Models for Software Package Relevant condition for software package organization is a search of

common features and statements in modeling of different objects. Unificated modules must be used where it is possible. All noted-above problems can be solved by the unique method of numerical integration of differential equations, namely, Adams-Fulton’s or Runge-Kutta’s method.

Subroutine RUNGE from Annex 3 which integrates the system )...,,,( 21

`nkk yyyxfy of ordinary differential equations by applying Runge-

Kutta’s method with automatic step selection is strongly efficient. The following parameters have to be assigned for program operation: the initial value of independent variable x; the initial values of desired functions y[k]; the order of system n; the procedure that calculates the right side of equations and array of derivatives z[k].

C-version of the translated Algol program [12] is given in Annex 3. The conception being laid into RUNGE complex is similar to that in ZEROIN. Parameters in the right sides of differential equations are independent data type and can be transmitted as pointers onto structure.

For reflecting systems being described (n=2), the corresponding parameters are presented in Table 3.1. The sign before multiplier in Table 3.1 may be arbitrary (sec above).

Table 3.1 System variant Variable Function

y1 Function

y2 z(1) z(2)

Remote zone: axially symmetric

system φ α r

sin)(sin)`(

lI

2

tan r

cylindrical system φ α r )(

)`(

lI

2

tan r

Proximate zone axially symmetric

system x φ r sin)(

)(l

xxE

2tan)1(

zr

cylindrical system x φ r

)()( lxE

2tan)1(

zr

154

3.2. METHODS AND ALGORITHMS OF SOLVING INVERSE PROBLEM FOR LENGTHY SOURCES

The Principles of Solution Since the light field of a point source can be represented in the form of

independent light tubes, the solution of inverse problem exists. This principle makes a foundation in vector method proposed by N. G. Boldyrev and A. A. Gershun [13] for designing the shape of specular surface. Inversion of operator, which calculates the intensity when the mirror shape is prescribed, leads to the equation with respect to Gaussian curvature (sec Ch. 2), i. e. to differential operator of the second order, that further can be reduced to a system of two first-order equations.

As for a source with finite dimensions, the light-tube structure becomes sufficiently complex, and reflected tube geometry can not be determined without knowing the properties of reflecting surface.

We analyse image structure for given direction ao by sending a set of parallel rays with a flat wave-front towards a reflector (Fig. 3.3a). The extreme points of a flashed zone (an image) in a meridian plane are determined by internal rays AoBo and A1B1 that are tangent to a source contour. Angle coordinates φ' and φ' of these points can be found, e. g. when appropriate formulas given in Ch. 2 are used.

We set an angle of observation a1 close to a0; aaa 01 (Fig. 3.3b). Transition to a1 > a0 results in image displacement

Fig. 3.3. Isogonal trajectories for a=ao (a) and a=a1 (b); and design of zero and consequent

zones of reflector.

155

by δφ>0 under accepted ray-path scheme. Situation is repeated for a2>a1. We call the straight lines )(

2)(

1 , kk HH containing all image points for given direction endaa as isogonal directions (isogones). Isogones form original tubes that do transfer light source images, but not luminous flux.

Hence, the luminous intensity curve of a reflector can be represented as a sum of zonal (virtual) candlepowers;

,)(1

)(

N

k

kaIaI

while ),()( )(

kk

k aIaI 0)()1( k

k aI , ...2,1k Algorithm of Composing Specified Luminous Intensity Curve By using found relations ,, )(

2)(

1kk

k HHa we can construct an algorithm of composing specified luminous intensity curve (LIC) J(a) through the points

naaa ,...,, 10 0210 10,5,0( aaa o ... in Fig. 3.4).

Step 1. To determine an extreme point of the zero-zone B1 in such a way that .0

)1(aJI

Step 2. To compute the zero-zone LIC at the points naaa ,...,, 10 and «the lack» of luminous intensity at the point 1a :

).()()( 1)1(

11 aIaJadI Step 3. To find the point B2 so that the virtual candlepower yeilds

).()( 11)1( adIaI

Step 4. To calculate the first-zone LIC at the points naaa ,...,, 10 and «the lack» of luminous intensity

.)()()(2

12

)(22

k

k aIaJadI

Cyclicity of calculations is obvious. The said method resembles the well-known method of interpolating

continuous function in terms of a system of orthogonal functions. We note, that known method of filling the LIC with zonal curves

156

Fig. 3.4. On algorithm of composing specified candlepower curve from set of zonal curves.

157

does not provide candlepower preservation during transition from one zone to another, bccause zones with constant angular width have been regarded there. In our ease we do not fix the zone angular dimension but prescribe the directions ak (isogone family). Flashed zones overlap with each other, so, that the point B1 of a new surface segment becomes the inner point of the first zone (Fig. 3.3b).

Construction of Mirror Shape We describe the method of constructing (synthesis) a surface from a family

of parabolas. Let the initial point Bo(xo, zo) be set (Fig. 3.3b). The straight line Ao Bo can be regarded as a line, along which the focus F0 of parabola runs, while the axis of parabola is in parallel with isogones H1, H2. Assigning «the lack» of intensity dI(ao) and using successive approximations, we find the point Fo of parabola B0B1 to satisfy the following inequality

100 )()(21

aIaI BB , where is the prescribed error.

For 01 aa the image extinguishes at point B0 and appears again at point `1B (the isogone H1), while the internal ray moves from point A0 to point A1.

Now we may assume that segment A'A1 is a light source, and A'B1 is the foci line of consequent parabola are B1B2 of the second zone.

Setting up dI(a1), we define focus point F1 in such a way that the following inequality is valid

)()( 1121aIaI BB .

The process is being continued until the complete image is constructed. However, the extension of parabolic segments may appear impossible if

the prescribed curve possesses sharp overfalls. Obviously, two solutions are possible within described method; within

elliptic class of reflectors when isogones turn into positive direction of angles a (see Fig. 3.3); and within hyperbolic class of reflectors when isogones turn into negative direction. The lack of luminous intensity is calculated in each cycle by inverse-ray method (see Ch. 2).

Since segment A1B1 remains fixed during the extention of a parabola segment, this procedure provides the smooth first-order connection of curves.

158

Cylindrical System V. D. Komissarov proved that the solution for cylindrical system could be

obtained in finite form [9]. We suppose to reproduce the LIC at the transversal plane. Let point B1 be the initial point of reflector profile or the succesive point where calculations were stopped. Starting from the condition that virtual luminous intensity has a prescribed value, we find parabola are B1B2. Obviously, it is sufficient to set a projection ∆H=H2-H1 (Fig. 3.5). We take an advantage of the fact that optical path-ways for the rays towards the front of arbitrary reflected wave, say C1C2, must be equal to each other. Consequently, the following path-ways have to be equal: A1B1+B1C1=L and A2B2+B2C2=L. From the triangle A1B1C1 we have

,)cos()( `111

`1111 OAOCaAABA

or ),(cos)](sin[)cot( 1

111111 aaRHaRBA (3.23)

whence )(tan)(cos 1

11

1111 aRaHBA , (3.23)

Fig. 3.5. Construction of zero-zone profile (point B2) for cylindrical source.

159

or

.)(cos

)tan(

11

111 a

aRH

BA

From the same triangle we obtain ),(cos)tan()tan( 1

11111

`111 aRaHaCACB

or

.)tan(

)(cos

1

11

111 a

aRH

CB

From the triangle A2B2C2 we find

),(cos])(sin[)cot(

);(cos)(21

2212

22

212

`22

`222

aHaRaRBAaOCOAAABA

whence ),tan()(cos 21

2222 2 aRaHBAl or

.)(cos

)tan(

21

2222 a

aRH

BA

Further ),(cos)tan()tan( 2

12222

`222 aRaHaCACB

or

.)tan(

)(cos

2

21

222 a

aRH

CB

Hence, we obtain )].tan()()[cos( 11

1111111 aaRHCBBAL (3.24)

Now we have an equation for determining parameters of point B2 ,)]tan()()[cos( 112

12 LaaRH (3.25)

where L1 is calculated from Eq. (3.24).

160

With the aid of Eq. (3.25) we find angle θ2 and then other elements:

.2

;90;/tan

;)()()sin(

2222

220

2

22

22

21

22

21

2

RLr

lRRHLRHLa

In order to pass over to the next parabola segment wc have to find the initial point X of the first isogone. Wc introduce the following denotions (Fig. 3.6):

φ, r are the polar coordinates of point X in the coordinate system having the centre at the point O;

t, are the polar coordinates in the local system having the centre at the point F;

12 is an angular dimension of the source viewed from the point F; ψ is an angular dimension of segment FX viewed from the point O; ψ' is an angle defining the direction of the ray A1X after reflection.

Fig. 3.6. Extension of consequent zone to zero-zone.

161

From the triangle OFX we have );cos(2 11

222 FOttFOr (3.26)

.sin)sin( 11 rFO (3.27) From the triangle OXA`

1 we obtain .)`sin( Rr (3.28)

We add parabola equation to Eqs. (3.26)-(3.28)

.cos1

)cos1(

1

11

FBt (3.29)

Equations (3.26)-(3.29) define the coordinates ,,, rt of the point X. Polar coordinate φ of unknown point can be found from equation

)sin(sin 11 tr , or

).sin()cos( 1112 tr (3.30) From Figs. 3.5 and 3.6 we find the constituent parameters of Eqs. (3.26)-

(3.30):

11211

21111111

,2/)(;2/)cot(

RBAFABAFB

(3.31)

The virtual luminous intensity of subsequent segment must ensure the following

.`)(`)(2XBreqvirt ahahh

The described method of designing specular surface by composing parabola segments provides smooth connection between curves up to their second derivatives. From this point of view the method is similar to that of Boldyrev.

162

Differential Equations for the Profile of Specular Surface Like any other evolutionary process, generally, the procedure of designing

the profile of specular surface, that ensures given candlepower or illuminance distribution, is described by a system of differential equations similar to those proposed by Boldyrev. We derive these equations for axially symmetrical system combined with a source stretched along the axis OZ (Fig. 3.7); it can be a tubular source, in particular [14].

The first stage of calculation The first segment, or the zero-zone of reflector, is determined by basing on

the condition of ensuring the prescribed luminous intensity Jao at the direction a0. For this segment the mirror surface equation holds true (Fig. 3.7a)

.2

tan 0adxdz

We rewrite it in another form:

.2

tan 0adzdx

dxdz

s

(3.32)

where linear coordinate zs is counted along the light source from system origin O, which is the centre of the source with the length of 2l.

Fig. 3.7. On deduction of differential equations for spccular surface: a - the first stage; b - the second stage.

163

From the same figure we obtain ;sincos 00 azaxH (3.33)

;sincos 00 adzadxdH (3.34) Furthermore, we can suppose a relation between image projection dH and

corresponding displacement dzs of a ray along the source: ,)( ssao dzzTkdH (3.35)

where kao is the amplification factor. The function T(zs) defines movement rate of internal ray along the source.

It characterizes the multivalencc of solution. For simplicity, we set up T(zs)=1. From Eqs. (3.33) and (3.34) we can find dx/dzs and substitute it into Eq. (3.32). After transformations we obtain

.2/)cos(2/)cos(

;2/)cos(2/)sin(

0

0

0

0

aak

dzdx

aak

dzdz

aos

aos

(3.36)

The right sides of Eq. (3.36) are the functions of x, z, zs, if we account the

following relationship

.arctanzz

xs

(3.37)

There exists the only solution of parametrical equations (3.36) under initial conditions x=x0, z=zo, and it can be found by applying numerical methods. The right sides of Eq. (3.36) can be reduced to parameters zs and z. Integrating Eq. (3.35), we obtain

),(0 lzkHH sao whence, accounting Eqs. (3.33) and (3.37),

.)(cos

sin)(arctanzza

azlzkHso

ossaoo

(3.38)

Equation (3.36) together with Eq. (3.38) defines the profile of the zero-zone for zs=-l, z=z0.

The value of kao can be found by successive approximations. In order to obtain the first approximation we integrate Eq. (3.35)

164

l

ls

H

HOao dzkdH ,

whence

.22 c

oao l

HlhHk

The kao-value can be estimated approximately if the image along α-direction is known. If α>rc/ro, where rc is the tube radius, and ro is the radius-vector of initial point, then according to theory of direct calculation (see Ch. 2) we obtain the expression for the flashed area

.sin

12a

rHS fl

Assuming source luminance to be Lo, we get

.4

sin

cco

aao lrL

aJk (3.39)

For α=0, we, obviously, have an assessment from below

,)sin(2 00max

0

rRlJk

cao (3.40)

where Rmax is a radius of exit aperture. Generally, internal rays corresponding to isogonal trajectories form a

caustic, the specific case of which is the focus (when parabolic zones are selected).

At the first stage of calculation by inverse-ray method we find the LIC for the zero-zone (compare with the case of cylindrical source), and this serves as initial approximation for the second stage.

The second stage On the second stage we add to already-designed surfacc segment another

one that reproduces the deficient of luminous intensity at given direction. Constructed in this way, the zone becomes initial for the next zone, and the calculation is being continued until the prescribed LIC is composed.

If a light source is substituted by elementary segment, the scheme

165

of ray tracing for added zone is similar to that of the zero-zone (Fig. 3.7b). This corresponds to discussed-above composition from paraboloid zones.

We have the following relationships [see Fig. 3.7b] ;sin dardzs (3.41)

;])()2/[( 2/122 lzdxr (3.42)

.2/tanclz

dx

(3.43)

Taking into account Eq. (3.35), we obtain (omitting index 0)

.2/)cos(2/)cos(

sin

;2/)cos(2/)sin(

sin

aark

dadx

aark

dadz

ao

ao

(3.44)

Equations (3.44) combined with Eqs. (3.42) and (3.43) give a solution under initial conditions x=x1, a=a1, z=z1 (where parameters x, a, z characterize the terminal point of the zero-zone).

Transition from α to α+dα gives the following deficiency of luminous intensity

,)]`()`([ daaIaJdIa (3.45) where I'(α) is a derivative of luminous intensity of reflector. Since the deficiency of intensity can not be negative, we get

).`()`( aIaJ (3.46) Condition (3.46) is similar to that for the kernel of Fredholm’s first-genus

integral equation. If, e. g. the kernel has a continuous derivative with respect to argument, then the right side of equation must contain the continuous derivative with respect to the same argument as well.

The peculiarities in solution of the system of equations are such, that we do not know the relationship k=k(a) beforehand. We can only prescribe it approximately in accordance with Eqs. (3.39) and (3.40)

)0(~ Jka for α=0 aJka sin)0(~ for α>0

166

Since the source possesses finite angular dimension , the construction of profile has to be terminated at the point αmax, where the positiveness condition of dI is violated [see Eq. (3.45)]. Smoothness of calculated profile and curvature sign constancy are the important properties of obtained solution; this result is ensured by the ray-tracing scheme being chosen.

3.3. CALCULATING SCHEMES FOR REFLECTOR OPTIMIZATION (A) In fixtures where extended radiating sources are applied the profile of

reflector has to be chosen in a way that flux losses associated with return of rcflcctcd rays back onto lamp bulb are reduced to minimum. Let a source be of elliptic form (sec Ch. 2). We demand that marginal (tangent) ray after reflection must follow the same direction. Consequently, a point of reflector circumscribes an evolvent of ellips. Evolvent equation for a curve under arbitrary parametrization r(t) is as follows

,)(`)(`

)`()(~ 2

2dttr

trtrtrr

or

.````

)`()(~

,````

)`()(~

22

22

22

22

dtzxzx

tztzz

dtzxzx

txtxx

(3.47)

Substituting ellips equation (see Ch. 2) into Eq. (3.47), we obtain

)]},,(),([sincos

cos{sin~

)]};,(),([sincos

sin{cos~

02222

02222

teEteEtba

Tataz

teEteEtbta

tatbx

(3.48)

where E(e, t) is the elliptic integral of the second genus; t0 is a parameter, the choice of which depends on initial point of evolvent.

167

If a source has a circular form (e=0), then evolvent equation lakes the form*

}.cos)({sin~};sin)({cos~00 ttttazttttax (3.49)

(B) It is a known fact that paraboloid makes the most concentralion of luminous flux emitted by sources of small angle dimensions. Light-tubes associated with such sources represent a system of concentric cones. Demanding reflected flux to be aimed at prescribed direction or towards fixed point, we impose certain conditions on normals at points of reflector; this leads to differential equations with known solutions - they are paraboloid and ellipsoid. For extended sources light-tubes are more perplexed. For a source of uniform luminance they comprise a family of hyperboloids with tangenls being the bisectrices of corresponding angles between radius-vectors connecting marginal points of a light source and a point of reflector (Fig. 3.8).

We demand a light-vector which is tangential to hyperbola to run in parallel to the axis after reflection.

It is easy to see that the similar scheme of ray-tracing corresponds to more dense packing of elementary maps on a plane near the point α=0. The bisecrix of an angle at the vertex forms an angle with the axis equal to (θ1+θ2)/2 (Fig. 3.8).

Obviously, the polar angle of normal is equal to (θ1+θ2)/4. Hence, the profile equation of the «optimal» reflector with elliptical source takes the following form [see Ch. 2]

.)()(

2arctan41tan 2222

bxaz

zxdxdz

(3.50)

If the source has a filament form of 2l length, then Eq. (3.50) yeilds [15]

.2arctan41tan 222

xlzzx

dxdz

(3.51)

* Reflectors with profiles described by Eq. (3.49) are used, in particular, in luminaires with fluorescent lamps.

Fig. 3.8. On construction of bisectroid.

168

Under prescribed initial conditions x=xo, z=zo, Eqs. (3.50) and (3.51) define the only curves lhat meet the requirement of maximum concentration along axial direction [16].

3.4. OPTIMIZATION OF FOCAL PARAMETER OF REFLECTOR In floodlight design a throat diameter 2a of rcflcctor and an exit-aperture

diameter 2b arc usually determined by constructive, thermal and lighting requirements. Under given constraints the choice of focal parameter p of paraboloid, that provides the maximal utilization factor of luminous flux, is difficult.

Let a source be placed at a parabola focus loeated at the centre of polar coordinate system (Fig. 3.9). Polar angles that define the throat and the margin of reflector are as follows

)]./()arccos[(

)];/()arccos[(2222

22220

bpbpapap

e

(3.52)

The flux falling onto reflector within angles φo and φe is a function of focal parameter

dIрФp

p

e

o

sin)(2)()(

)( , (3.53)

where I(φ) is candlepowcr curve of a source. The necessary condition for reaching maximum of utilization factor yeilds

.0dpdФ

(3.54)

Fig. 3.9. Parameters of paraboloid reflector.

169

Differentiating Eq. (3.53), we get

0]sin)(sin)([2 dp

dJdpdJ

dpdФ o

ooc

cc . (3.55)

It follows from Eq. (3.52) that

,sin'sin

ecoo

dpd

dpd

(3.56)

hence, the condition (3.54) can be transformed

ecII 20

20 sin)(sin)( . (3.57)

Equation (3.57) can be regarded as an equation in p, if we express φ0=φ0(p) and φe=φe(p) explicitly from Eq. (3.56)

We add to Eq. (3.57) one more equation expressing the fact that the points corresponding to φ0 and φe lie at the same parabola

.sin

cos1sin

cos1

e

e

o

o ba

(3.58)

If Eqs. (3.57) and (3.58) have a single solution p0 that meets the condition

,02

2

podp

Фd

then p0 defines the desired parabola. If the function I(φ) is symmetrical with respect to φ=π/2, it becomes easier

to solve Eq. (3.57). For analytical calculations the following representation of I(φ) is suitable

.sin)(0

mN

mmgI

Then Eq. (3.57) takes the following form

0)sin(sin 20

2

0

e

mmN

mmg . (3.59)

If all gm are nonnegative, then all solutions of Eq. (3.58) meet the condition 0coscos 0 e .

170

Fig. 3.10. Relation between flux Ф falling onto reflector and focal parameter p: p1 = a,

;, 2 bpabpo 1 – I(φ)-cosφ;

2 – I(φ)-cosφ-1; 3 – I(φ)-sinφ.

This equation yeilds the following solution .abpo (3.60)

If all gm are nonnegative, then Eq. (3.58) can have different solutions, but for majority of «reasonable» curves I(φ) the solution being found is the global maximum (Fig. 3.10).

Obtained results are applicable when angular dimensions of a source do not exceed the angle a , within which flux concentration is required. In case when this condition is violated, there is a chance to choose an optimal in this sense parameter p by considering relationships F=F(p) and )( p simultaneously.

When overall dimensions of reflector are specified, the optimization of paraboloid focal parameter enables to increase utilization factor of luminous flux. Conversely, when utilization factor is prescribed, it allows to decrease the overall dimensions of reflector.

3.5. CALCULATION OF MILLING-CUTTER TRAJECTORY

If a surface is treated by cutting instrument with round head, the centre of instrument moves along parallel surface which is apart from the former at a distance equal to milling-cutter radius R. Surface parallax also has to be taken into account when specifying tolerance on thickness of metal sheet.

If surface is set in parametric form, i. e. r=r(u, v), then the corresponding surface which is displaced at a distance d along normal n(u, v) can be described as follows

).,(),(` vundvurr (3.61) If G and G' are the matrices of the first fundamental forms of said surfaces,

then

171

ndkdkGnG )1)(1(` 11 (3.62) where k1 and k2 are the principal curvatures of initial surface. The radii of the principal curvatures in biased surface [17] are

);1( 11`1 dkkk ).1( 22

`2 dkkk (3.63)

If surface is set in terms of its sections then a software based on spline representation of curves (see Annex) is suitable for calculating milling-cutter trajectory. Number of points Ne or point spacing h can be adopted as the input data in this case.

Algorithm for calculating milling-cutter trajectory is as follows: (1) to introduce arrays of section coordinates X(I), Y(I), NI ,1 ; (2) to prepare spline coefficients for input arrays; (3) to find the number of layout points (for given step) N=(X\N] – X[l])/(h

+ 1); (4) for each interpolated point (x, y) in a cycle with respect to Nke ,1 to

do the following: (a) to find the normal n = [nx, ny ]T; (b) to calculate the coordinates of parallax surface

;` xRnxx .` yRnyy The calculated points define linear segments of milling-cutter trajectory.

Nevertheless, the milling-cutter radius, obviously, must not excced the minimal curvature radius of a surface.

Appropriate reference list on computer-aided manufacture of surfaces with regard of tolerances and other restrictions can be found in [18].

172

REFERENCES

1. Boldyrev N.G. About Calculation or Asymmetrical Specular Reflectors //Svetotekhnika, 1932, № 7, p. 7-8.

2. Komissarov V.D. The Foundations of Calculating Specular Prismatic Fittings // Trudy VEI, Issue 43, 1941, p. 6-61.

3. Schruben J.S. Formulation of a Reflector-Design Problem for a Lighting Fixture // J.Opt.Soc.Am. 1972, Vol. 62, № 12, p. 1498-1501.

4. Schruben J.S. Analysis of Rotationally Symmetric Reflectors for Illuminating Systems // J.Opt.Soc.Am. 1974, Vol. 64, N9 I, p.55-58.

5. Keller J.B. The Inverse Scattering Problem in Geometrical Optics and the Design of Reflectors // JRE Trans, of Antennas and Propagation, Vol. AP-7, Apr. 1959, № 2, p. 146-149.

6. Kinber B.B. The Solution of Inverse Problem in Geometrical Acoustic // Acoustic Journal, 1955, Vol. I, № 3, p. 221-225.

7. Cornbleel S. Microwave Optics. Academic Press. New York — San Francisco — London, 1976.

8. Wolber W. Berechnung vir Reflektoren fur beliebigc Lichtverteilungen // Lichttechnik, 1970, Vol. 22, № 12.

9. Komissarov V.D. On Calculation of Cylindric Specular Reflectors for Incandescent Lamps with Tungsten-Iodide Cycle // Materialy IX nauchno-tekhnich. konferentsii po osvetit. priboram, Ternopol', May 1969, M.: Informelektro, 1969, p. 69-86.

10. Stanioch W. Calculation of Reflecting and Refracting Optical Elements // Light. Res & Tcchn. 1991, Vol. 3, № 3, p. 145-149.

11. Rymov A.I., Skoblova V.I. Melhod of Calculating Reflector Profile in Imitators of Solar Radiation // Svetotekhnika, 1978, № 3, p. 3-5.

12. The Library of Algorithms lb-50b. Reference Book. M.: Sovetskoe Radio, 1975.

13. Boldyrev N.G., Gershun A.A. Vector Method of Calculating Symmetrical Specular Reflectors // Svetotekhnika, 1936, № 1, p. 7-9.

14. Korobko A.A., Kusch O.K. Construction of Mirrow Surface of a Luminaire with Lengthy Light Source // Svetotekhnika, 1982, №3, p. 3-6.

15. Inventor's Certificate 1227908, USSR, MKI F21 V7/04. Irradiating Installation // Discoveries and Inventions, 1986, № 16.

16. Gehring A.P., Holten P.A.J. A System Approach to Incandescent Reflector Lamp Development // J. of IES, Summer 1990.

17. Willmore T.J. Differential Geometry. Oxford University Press, 1958. 18. Faux I.D., Pratt J. Computational Geometry for Design and

Manufacture. Ellis Horwood Ltd., 1979.

173

ANNEX 1 D.Yu. Chepelevskii has composed program texts given below.

PROGRAM COMPLEX ZEROIN

***************************************************************************** * MATM_EXT.H Header file for declarations of mathematical routines * * and macros. * * ************************************************************************* #ifndef DEFMATH_EXT /* Prevent second reading of */ #define DEF_MATH_EXT 1 /* these definitions. */ #include ath.h /*______________________Definitions_________________________ */

/* 2*Pi. */ #define M_2PI 6.28318530717958647692

/* Pi. */ #define M_PI 3.14159265358979323846

/* Pi/2. */ #define M_PI_2 1.57079632679489661923 #define BEGIN { /* Left bracket. */ #define END } /* Right bracket. */ /*_______Constants for use with AllcCurv (the «option» value)____ */ # define ALLOC_ARG 1 #define ALLOC_FUN 2 #define ALLOC_B 4 #define ALLOC_C 8 #define ALLOC_D 16 #define ALLOC_BCD (ALLOC_B I ALLOC_C I ALLOC_D) /*________Constants for type of curve interpolation______________ */ #define INTER_LINEAR 0 /* Linear interpolation. */ #define INTER_SPLINE 1 /* Spline interpolation. */ /*__________________Macros______________________________ */

/* Conversion from radians to degrees. */ #define DEG(x) ( (x)*180./M_PI )

/* Conversion from degrees to radians. */ #define RAD(x) ( (x)*M_PI/180. )

/* Whether the value is odd. */ #define ODD(x) ( (x) & 0x0001 )

/* Whether the value is even. */ #define EVEN(x) ( !ODD(x) )

/* Choice of maximum value. */ #define MAX(x,y) ( ( (x) > (y) ) ? (x) : (y) )

/* Choice of minimum value. */ #define MIN(x,y) ( ( (x) < (y) ) ? (x) : (y) )

/* Square of value. */ #define SQR(x) ( (x)*(x) )

/* Cube of value. */ #define CUBE(x) ( (x)*(x)*(x) )

174

/* Sign of value. Returns -1 if x 0, */ /* 0 if x = 0 or I if x > 0. */

#define SIGN(x) ( ((x) >0) ? (1) : ( ((x) < 0) ? (-1) : (0) ) ) /* Inversion of two values. */

#define /* INVERS (a,b, tmp) tmp = a; a = b; b = tmp /*______________________________Definitions of data types */ typedef struct /* Data structure of coefficients arrays */ { /*for spline interpolation. */ int init_cfs; /*Flag of coefficients initiation. unsigned last_knot; /*The knot number from last call. */

/* If coefficients are initiated, flag */ /* is equal to I. */

Double *b; /* Pointer to the array. */ Double *c; /* Pointer to the array. */ Double *d; /* Pointer to the array. */ } SPLN_COEFS; */ typedef struct /* Data structure for curve description. */ { */ Char * name; /*Name of the curve. */ Int option; /*Bitwise OR-ing of the following: */

/*ALLOC_ARG — allocate arguments array */ /*ALLOC_FUN — allocate function value array */ /*ALLOC_BCD — allocate spline coefficients /*arrays. */

Int_typeinter; /*Type of interpolation. */ Unsigned nmb; /*Number of points on the curve. */ Double *arg; /*Array of curve arguments. */ Double *fun; /*ray of curve function values. */ SPLN_COEFS cfs; /* Structure of spline interpolation coefficients. */ } CURVE; /* */

/* Type of working function for */ /* use in the function «Zeroin». */

typedef double FUN_ZEROIN( double arg, void *prm ); typedef struct /* Data structure for calculating the */ { /* root in the function «Zeroin». */ FUN ZEROIN *fun; /* Pointer to the working function used */

/* in the function «Zeroin» for root */ /* calculation */

Void *prm; /* Pointer to the parameters structure */ /* used in the working function. */

} ZEROIN; /* */ /* ___________________________Function declarations_________ */ void InitCurv ( CURVE* ); /* Initializes pointers to the arrays. */ int AllcCurv ( CURVE* ) ; /* Allocates arrays for curve. */ void FreeCurv ( CURVE* ) ; /* Frees arrays allocated with AllcCurv. */ unsigned BnrSrch (double, /*Finds the left knot. */

CURVE* ); /* */ void Line ( CURVE* ) ; /*Initializes coefficients for linear */ /*interpolation. */

175

void Spline ( CURVE * ); /* Initializes coefficients for spline */ /* interpolation. */ double Seval ( double, /* Chooses function value from the curve */ CURVE *, /* using spline or linear interpolation. */

double *, /* Finds values of first and second */ double * ); /* derivative. */

void Quadr (CURVE *, /* Integrates the function. */ double [] ); /* */

int Zeroin (double, /* Calculates argument value when */ double, /*function is equal to zero. */ ZEROIN *, /* */ double, /* */ double * ); /* */

#endif /* Ends «#ifndef DEF_MATII_EXT» */ #include «malh_ext.h» / ******************************************************************************** * Name Zeroin - finds the root of equation with one variable. * * * * Synopsis error - Zeroin( ax,bx, f, tol, root ); * * int error - Code of error; * * double ax,bx - Left and right borders of initial interval. * * ZEROIN *f - Pointer to the structure with pointer to the * * working function (that allows to calculate * * f(x) for any point on the interval [ax,bx]) * * and pointer to the parameters structure used * * in the working function. * * double tol — The desirable length of indefinite interval. * * double *root} — Pointer to the root value. * * * * Description This function is intended for calculation of real zero of * * function f(x) = 0. Zcroin combines reliability of bisection * * method and speed of chord method. It decreases the length * * of indefinite interval until |b-a| < tol + 4*eps*fabs(b). * * After this b is equalized to value of root. * * * * Zeroin represents C-realization of Forsythe’s fortran program. * * * * Returns error — Error code is equal to -1 or 1 if function * * values on the borders of the interval have the same sign * * and the root cannot be found. Otherwise 0 is returned. * ******************************************************************************** / int Zeroin ( double ax, double bx, ZEROIN *f, double tol, double *root ) BEGIN double a,b,c,d,e; double eps=1., /* Relative machine accuracy. */ toll-2.; /* High border of error. */

176

double fa,fb,fc; /* Va!ues of the function. */ double xm,p,q,r,s;

/* Calculate relative machine accuracy. */ while( toll > 1.) { eps *= 0.5; toll - 1. + eps; }

/* Find function values on the borders. */ a = ax; b = bx; fa = (f->fun)(a,f->prm); fb = (f->fun)(b,f->prm);

/* If function values on the borders */ /* less then permitted error, root is */ /* equated to the value of interval */ /* border (a,b). */

if( fabs(fa) < tol ) { *root = a; return( 0 ); } if( fabs(fb) < tol ) { *root - b; return( 0 ); }

/* If function has the same sign on the */ /* borders return its sign. */

if( SIGN(fa)*SIGN(fb) > 0 ) return( SIGN(fa) ); /*__________________________________Main loop___________________________ */ do {

c = a; fc = fa; d = b - a; e = d; Ll: if( fabs(fc) < fabs(fb) ) {

a = b; b = c; c - a; fa = fb; fb - fc; fc = fa;

} /* Check up the permitted error */

toll - 2.0*eps*fabs(b) + 0.5*tol; xm - 0.5*(c-b); if( (fabs(xm) <= toll) I I (fb — 0.0) ) { *root = b; return( 0 );

} if( (fabs(e) >= toll) && (fabs(fa) > fabs(fb)) ) {

/* Reverse the square interpolation. */ q = fa/fc; r = fb/fc; s = fb/fa; p = s*( 2.0*xm*q*(q-r)-(b-a)*(r-1.0) ); q = (q - 1.0)*(r - 1.0)*(s - 1.0); if (a- c ) { /* Linear interpolation. */ s * fb/fa; p = 2.0*xm*s; q = 1.0 - s; } if( p > 0.0 ) q *= -1.; p =* fabs(p);

/* Whether the interpolation is */ /* acceptable. */

if( (2.0*p) < (3.0*xm*q-fabs(toll*q)) && (p<fabs(0.5*q*e)) ) { e = d; d = p/q; goto L2; } } d = xm; e = d; /* Bisection. */

L2: a = b; fa - fb; if( fabs(d) > toll ) b += d; else b += (xm > 0) ? fabs(toll) : (-fabs(toll)); fb = (f->fun)(b,f->prm);

} while( (fb*(fc/fabs(fc))) > 0.0 ); goto Ll; END

177

ANNEX 2 PROGRAM COMPLEX SPLINE

#include «math_ext.h» ******************************************************************************** * Name Line — initializes coefficients for linear interpolation. * * * * Synopsis void Line( crv ); * * CURVK *crv — Pointer to the structure of curve arrays. * * * * Description This function calculates coefficients for linear interpolation: * * B and C. * * * * Arguments of the curve must be in strictly increasing order. * * * * Returns Nothing. * ******************************************************************************** / /*________________________Internal definitions ______________________ */ #define N crv->nmb /* The number of points on the curve. */ #define X crv->arg /* Array of curve arguments. */ #define Y crv->fun /* Array of curve function values. */ #define B crv->cfs.b /* Arrays of interpolation coefficients. */ #define C crv->cfs.c /* */ #define D crv->cfs.d /* */ void Line ( CURVE *crv ) BEGIN unsigned i;

/* Whether coefficients were initialized. */ if( crv->cfs.init_cfs | | N < 2 ) return; for( i=0; i < N-1; I++) {

B[I]=(Y[i+1]-Y[i])/(X[i+l]-X[i]); C[i] - D[i]= 0;

} B[N-1] = B [N-2]; C[N-I] – D[N-1] = 0;

/* Set flag that coefficients are */ crv->cfs.init_cfs = 1; /* initialized. */ END

178

#include «math_ext.h» / ********************************************************************************** * Name Spline — initializes coefficients for spline interpolation. * * * * Synopsis void Spline( crv ); * * CURVE *crv — Pointer to the structure of curve arrays. * * * * Description This function calculates coefficients for cubic spline * * interpolation: B,C and D. * * * * Arguments of the curve must be in strictly increasing order. * * * * Spline represents C-realization of Forsythe's fortran program. * * * * Returns Nothing. * / ********************************************************************************** /*______________Internal definitions_____________________ */ #define N crv->nmb /* The number of points on the curve. */ #define X crv->arg /* Array of curve arguments. */ #define Y crv->fun /* Array of curve function values. */ #define B crv->cfs.b /* Arrays of spline coefficients. */ #define C crv->cfs.c /* */ #define D crv->cfs.d /* */ void Spline( CURVE *crv ) BEGIN unsigned ib,i; /* Counters. */ double t; /* Variable for temporary use. */

/* Whether coefficients were initialized */ if( crv->cfs.init_cfs I I N < 2 ) return; if( N !=2 ) { /* Construction of three diagonal system. */

D[0] - X[1] - X[0]; C[1] - (Y[1]-Y[0]) / D[0]; for( i=l; i < N-1; i++) {

D [i]=X[i+l] - X[i]; B [i]=2 * (D[i-1]+D[i]); C[i+1] - (Y[i+l]-Y[i]) / D[i]; C[i] = C[i+1] - C[i];

} /* Border conditions. */

B[0] - -D [0]; B[N-1] * -D[N-2]; C[0] - C[N-1] = 0.;

if(N !=3 ) {

C[0] – C[2] / (X[3]-X [1]) - C[1] / (X[2]-X[0]); C[N-1] = C[N-2] / (X [N-1]-X [N-3]) – C[N-3] / (X[N-2]-X[N-4]); C[0] = C[0] * D[0] * D[0] / (X[3] -X[0]);

179

C[N-1] *= -D[N-2] * D[N-2] / (X[N-l]-X[N-4]); }

/* Direct passage. */ for(i=1; i<N; i++ ) {

t=D[i-1]/B[i-I]; B[i]-= t * D[i-1]; C[i]-= t * C[i-1];

} C[N-1] /= B[N-1];

/* Indirect substitute. */ for( ib=0; ib < N-1; ib++) { i = N - 2 - ib; C[i] = (C[i] - D[i] * C[i+1]) / B[i]; } B[N-1] - (Y[N-I]-Y[N-2]) / D[N-2]+D[N-2] * (C[N-2]+2. * C[N-1]);

/* Calculation of polynomial */ /* coefficients. */

for (i=0; i<N-l; i++ ) { B[i] = (Y[i+1] - Y[i]) / D[i] — D[i] * (C[i+1]+2.*C[i]); D[i] = (C[i+l]-C[i]) / D[i]; C[i] = 3*C[i];

} C[N-1] = 3. * C[N-1]; D[N-1] = D[N-2];

} else {

B[0] = (Y[1]-Y[0]) / (X[1]-X[0]); C[0] = D[0] = C[1] = D[1]=0.; B[1] = B[0];

} /* Set flag that coefficients are /*

crv->cfs.init_cfs - 1; /* initialized. /* END

180

#include «math_ext.h» ******************************************************************************** * Name BnrSrch - finds the left knot of the interval. * * * * Synopsis last_knot - BnrSrch( u, crv ); * * unsigned 1ast_knot - The knot number from last call. * * double u - Argument value where function is * * calculated. * * CURVE *crv - Pointer to the structure of curve arrays. * * * * Description This function finds the number of left knot of interval where * * spline is calculated. * * * * Returns The number of left knot. * ******************************************************************************** / /* ___________________________Internal definition__________________________ */ #define N crv->nmb /* Thc number of points on the curve. */ #define X crv->arg /* Array of curve arguments. */ unsigned BnrSrch( double u, CURVE *crv ) BEGIN unsigned i=crv->cfs.last_knot; /* Number of the left knot of the */ /* interval where spline is calculated. */ unsigned j,k; /* Counters. */ if( i >= N-I ) i=0;

/* If u doesn’t belong to the interval */ /* calculated in the previous call, use */ /* the binary search */

if((u<X[i]) | | ( u > X[i+1] ) ) { i = 0; j = N; do { /* The binary search. */ k = (i+j) >> 1; if(u < X[k] ) j - k; else i = k; } while( j > i+1);

} return( crv->cfs.last_knot=i); END

181

#include «math_ext.h» /******************************************************************************** * Name Seval — calculates value of function, first and second * * derivative at the point u. * * Synopsis fun_value = Seval ( u, crv, drvl, drv2 ); * * double fun_value — Interpolation value at the point — u. * * double u — Argument value where function is * * calculated. * * CURVE *crv — Pointer to the structure of curve arrays. * * double *drvl — Value of first derivative. * * If value of first derivative isn’t * * necessary drvl can be equal to NULL. * * double *drv2 — Value of second derivative. * * If value of second derivative isn’t * * necessary drv2 can be equal to NULL. * * Description This function calculates cubic spline or linear value at the * * point u using interpolation coefficients B, C, D. Before using * * interpolation coefficients must be initialized in the function * * Spline or Line (depends on type of interpolation). Seval also * * calculates values of first and second derivative at the point u. * * Seval represents C-realization of Forsythe's fortran program. * * Returns Interpolation value at the point u. * ******************************************************************************** /*_______________________Internal definitions_____________________ */ #define N crv->nmb /* The number of points on the curve. */ #define X crv->arg /* Array of curve arguments. */ #define Y crv->fun /* Array of curve function values. */ #define B crv->cfs.b /* Arrays of spline coefficients. */ #define C crv->cfs.c /* */ #define D crv->cfs.d /* */ double SevaK double u, CURVE *crv, double *drvl, double *drv2 ) BEGIN unsigned i; /* Counter. */ double dx;

/* Initialize interpolation coefficients. */ if( !crv->cfs.init_cfs ) {

if( crv->type_inter = INTER_LINEAR ) Line( crv ); else if( crv->type_inter = INTER_SPLINE ) Spline( crv );

} i = BnrSrch( u, crv ); /* Find the nearest knot. */ dx = u-X[i];

/* First derivative. */ if( drvl ) *drvl = B[i] + dx*( 2.*C[i] + 3.*dx*D[i]);

/* Second derivative. */ if( drv2 ) *drv2 = 2.*C[i] + 6.*dx*D[i];

/* Function value. */ return( Y[i] + dx*( B[i] + dx*( C[i]+dx*D[i] ) ) ); END

182

ANNEX 3 PROGRAM COMPLEX RUNGE.

/********************************************************************************* * * * RUNGE.H Header file for Runge-Kutta program complex. * * * ********************************************************************************** #ifndef DEF_RUNGE /* Prevent second reading of */ #define DEF_RUNGE 1 /* these definitions. */ /*__________________________Definitions____________________________ */ #define BEGIN { /* Left bracket */ #define END } /* Right bracket. */ #define NDE 5 /* Maximum number of differential */

/* equations. */ #define MIN_ORD_DBL -308 /* Order of minimal positive «doublc» */

/* value. */ /*_____________________Definitions of data types________________________ */

/* Type of working function calculating */ /* values of right parts of differential */ /* equations system for use in the */ /* function «Runge». */

typedef void FUN_RUNGE( double, double [ ], double [] void * ); typedef struct /* Data structure for use in the */ { /* Runge-Kutta program complex. */ int nmb_eq; /* Order of system of differential */

/* equations. */ FUN_RUNGE *fun; /* Pointer to the working function */

/* calculating values of right parts. */ void *prm; /* Pointer to the parameters structure */

/* used in the working function. */ } RUNGE; /* */ /*__________________________Function declarations __________________________ */ double Compa (double, /* Calculates absolute value of mantissas */

double, /* difference. */ double ); /* */

int Expon (double ); /* Calculates decimal order of floating */ /* point value. */

void RklStep (doubIe, /* Integrates system of equations on the */ double [ ], /* single step. */ RUNGE *, /* */ double, /* */ double *, /* */ double [ ] ); /* */

void Runge ( double, /* Integrates system of differential */

183

double | |, /* equations. */ RUNGE *, /* */ int, /* */ double, /* */ double | |, /* */ double, /* */ double ); /* */ #endif /* Ends «#ifndef DEF_RUNGE» */ #include <math.h> #include «runge.h» / ******************************************************************************** * Name Expon — calculates decimal order of a variable * * with floating point. * * Synopsis order = Expon( x ); * * int order — Decimal order of value x. * * double x — Floating point variable. * * Description This function is included in the Runge-Kutta program complex. * * When x <> 0, Expon calculates decimal order of value x. * * If x = 0, function returns order of minimal positive «double» * * value. * * Returns Decimal order of floating point variable for use in the * * function Compa. * ******************************************************************************** * int Expon( double x ) BEGIN if( x ) return( (int) (0.4342944819 * log(fabs(x))) + 1 ); return ( MIN_ORD_DBL ); END

184

#include «math_ext.h» #include «runge.h» / ******************************************************************************** * Name Compa — calculates the absolute value of mantissa * * difference * * Synopsis double Compa( a, b, c ); * * double a,b,c — Arbitrary values with floating point. * * * * Description This function is included in the Runge-Kutta program complex. * * It calculates absolute value of mantissa difference between * * the values a and b when their orders are equated to the * * largest order of v9h1es a,b and c. * * * * Returns Absolute value of mantissa difference between variables a and b. * ******************************************************************************** / double Compa( double a, double b, double c ) BEGIN int ae,be,ce; /* Decimal orders of values a,b and c. */ ae = Expon( a ); /* Decimal order of value a. */ be = Expon( b ); /* Decimal order of value b. */ ce = Expon( c ); /* Decimal order of value c. */ ae= MAX( MAX(ae,be), ce ); /* Choose maximal order. */ return (fabs(a-b)/pow(10.,ae) ); /* Calculate mantissa difference. */ END

185

#include «runge.h» /******************************************************************************** * Name RklSlcp - integrates system of equations on the * * Synopsis void RklStep (x,y,f,h,xh,yh) * * double x - Start point of integration. * * double y [] - The vector of initial values of sought functions. * * RUNGE *f - Pointer to the data structure with pointer * * to the working function calculating values of right parts of * * differential equations system and pointer to the parameters * * structure used in the working function. * * double h - Step of integration. * * double *xh - Pointer to the stop integration point. * * double yh[] - Vector of solutions at the point *xh. * * * * Description This function integrates system of differential equations on the single step. * * RklStep is used in the Runge-Kutta program complex. * * Returns Nothing * ******************************************************************************** Void RklStep(double x, double y[], RUNGE *f, double h,

double *xh, double yh[] ) BEGIN Int k,j; Double w[NDE],

a[5], z[NDE];

a[0] = a[l] = a[4] = 0.5*h; a[2] = a[3] = h; *xh - x; for( k=0; k<f - > nmb_eq; k++ ) {yh[k] - w[k] - y[k]; } for( j=0; j; j++ ) {

/* Right part */ (f->fun)( *xh, w, z, f->prm ); *xh - x + a[j]; for( k=0; k < f- > nmb_eq; k++ ) { yh[k] += a[j+l]*z[k]/3.; w[k] - y[k] + a[j]*z[k]; }

} /* «for(j)» */ END

186

#include < stdio.h > #include < process.h > #include «runge.h» / **************************************************************************** * Name Runge — integratessyslem of differential equations. * * * * Synopsis void Runge( x,y, f, prim, xfin, yfin, eps, eta ); * * double x — Start point of integration. * * double y[] — The vector of initial values of sought * * functions. * * RUNGE *f — Pointer to the data structure with pointer * * to the working function calculating values of * * right parts of differential equations system * * and pointer to the parameters structure used * * in the working function. * * int prim — Start parameter. It is equal to 1 on the first * * call. Then it is equal to 0. * * double xfin — Stop integration point. * * double yfin[] — Vector of solutions at the point xfin. * * double eps — Relative error of Runge-Kutta method. * * double eta — Absolute error of Runge-Kutta method. * * * Description This function is included in the Runge-Kutta program complex. * * Runge integrates system of differential equations using * * automatic choice of integration step. * * Returns Nothing. * ***************************************************************************** void Runge ( double x, double y[], RUNGE *f, int prim, double xfin, double yfin[], double eps, double eta ) BEGIN int i,k; /* Counters. */ int out; /* Flag for return. */ static int ss; /* Parameter for acceleration of */ /* calculation. */ static double hs; /* Start value of integration step. */ double h; /* Current step of integration. */ double yl [NDE], /* Arrays for estimation of error of */ y2[NDE], /* sought functions */ y3[NDE]; /* */ double xl, x2, x3; /* Intermediate points where yl[],y2[], */ /* y3[] calculated. */ if( prim ) { /* The first call. */ if(f->nmb_eq NDE ) { /* Check the number of equations. */ /* Print message. */ prinlf(“RUNGE: The number of equtions more than %i!\n",NDE ); exit (1); /* Terminate program. */

} h=xfin- x; ss = 0; /* Calculate initial step. */ }

187

else { h = hs; out = 0; } /* Restore step from the previous call. */ /* Main Ioop */

for( ;; ) { if( ( ( x + 2.01*h - xfin ) > 0 ) = ( h > 0 ) ) { hs = h; /* Remember step to calculate nexl point. */ out = 1; /* Set flag of return. */ h = 0.5*( xfin - x );

} /* Calculation wilh double step. */

RklStep( x, y, f, 2.*h, &xl, yl ); lab:

RkIStep( x, y, f, h, &x2,y2 ); RklStep( x2,y2, f, h, &x3,y3 ); for( k*=0; k < f - >nmb_eq; k++ ) {

/* Check condition of approach accuracy. */ if( Compai(yl[k], y3[k], eta ) eps ) { h *=0.5; out=0; xl = x2; for(i=0; i<f - > mb_eq; i++ ) yl [i] = y2[i]; goto lab;

} } x = x3; if( out ) {

for (k=0; k<f - > nmb_eq; k++ ) yfin[k] - y3[k]; return;

} for(k=0; k<f->nmb_eq; k++) y[k] = y3[k];

/* AcceIaration of calculation */ /* (increasing of step) after 5 */ /* repetitions. */

if (s=5) {ss=0; h*=2.;} else ++ss;

} /* «for ( ;; )» */ END

188

ANNEX 4

EXAMPLE FOR TESTING PROGRAM Program complex SHAPE is developed in order to calculate specular

reflectors with point sources. It is applicable for calculation of two types of reflectors; rotationally symmetric, and cylindrical. Both reflector types are defined by their profile-section. Program SHAPE uses program RUNGE (Annex 3) to calculate a reflector profile.

To debug programs similar to SHAPE we present the data on reflector which reproduce the following illuminance disstribution

),)/(1()))2/((1(4)( 222 HxfxxE where f and H are the parameters.

Obviuosly, the desired profile is a parabola with a focal distance f, while an isotropic light source is disposed at the height H above an area being lit. Appying formulas from Ch. 1, one can varify that paraboloid produces in the ccntre the illuminance four times greater than a flat mirror.

Assuming 2f=H=lm and having normalized the function E(x) to a maximim, we obtain

}.)1()1(4){5/1()( 1222 xxxE We also set the following: (1) the acceptance angle of reflector is equal to π/2; (2) the radius of lit circle is equal to l m; (3) the reflectance factor is equal to 1. We calculate luminous fluxes: (1) the luminous flux falling onto reflector

2/

0

;0.1sin1

dFs

(2) the direct flux falling on the target area

2928932188.02/11)1(2/31

0

2

dxxxFd ;

(3) the flux within prescribed curve E(x)

189

.2585786439.0)1)(5/1(

)1()1(4)5/1(1

0

2/3221

0

2

d

e

f

dxxxdxxxF

From the balance of fluxes we get

0.5/ eds FFF . SAMPLE OF REFLECTOR CALCULATION FOR PARABOLIC CASE 1. INPUT DATA. Luminous intensity distribution of the light source

N FI [deg] I [r.u.] 1 0.000 1.0000 2 20.000 1.0000 3 40.000 1.0000 4 60.000 1.0000 5 80.000 1.0000 6 100.000 1.0000 7 120.000 1.0000 8 140.000 1.0000 9 160.000 1.0000

10 180.000 1.0000 Illuminance distribution on the working plane

N X [m] E[r.u.] 1 0.0000 5.00000 2 0.0204 4.99605 3 0.0408 4.98421 4 0.0612 4.96458 5 0.0816 4.93730 6 0.1020 4.90257

190

N X [m] E[r.u.] 7 0.1224 4.86062 8 0.1429 4.81175 9 0.1633 4.75630 10 0.1837 4.69463 11 0.2041 4.62715 12 0.2245 4.55429 13 0.2449 4.47649 14 0.2653 4.39423 15 0.2857 4.30797 16 0.3061 4.21818 17 0.3265 4.12535 18 0.3469 4.02995 19 0.3673 3.93242 20 0.3878 3.83321 21 0.4082 3.73276 22 0.4286 3.63145 23 0.4490 3.52969 24 0.4694 3.42782 25 0.4898 3.32618 26 0.5102 3.22508 27 0.5306 3.12479 28 0.5510 3.02557 29 0.5714 2.92766 30 0.5918 2.83124 31 0.6122 2.73651 32 0.6327 2.64361 33 0.6531 2.55267 34 0.6735 2.46381 35 0.6939 2.37712 36 0.7143 2.29266 37 0.7347 2.21049 38 0.7551 2.13066 39 0.7755 2.05319 40 0.7959 1.97809 41 0.8163 1.90535 42 0.8367 1.83499 43 0.8571 1.76696

191

N X [m] E[r.u.] 44 0.8776 1.70126 45 0.8980 1.63784 46 0.9184 1.57667 47 0.9388 1.51771 48 0.9592 1.46090 49 0.9796 1.40620 50 1.0000 1.35355

Reflectance 1.0 Initial angle 0.0 [deg] Initial point 0.0 [m ] Initial radius 500.0 [mm] Final angle 90.0 [deg] Final point 1.0 [m] Height of the light center 1.0 [m] 2. POLAR COORDINATES OF REFLECTOR. ANGLE OF

REFLECTED RAY WITH SYMMETRY AXIS.

N FI [deg] ALFA [deg] R [mm ] REL. ERROR [%]

1 0.00 0.00000 500.00000 0.000000 2 2.00 0.00000 500.15234 0.000000 3 4.00 -0.00000 500.60973 -0.000000 4 6.00 -0.00000 501.37329 0.000000 5 8.00 -0.00000 502.44488 0.000000 6 10.00 -0.00000 503.82713 -0.000001 7 12.00 -0.00000 505.52345 -0.000000 8 14.00 -0.00000 507.53802 -0.000001 9 16.00 -0.00000 509.87586 -0.000001

10 18.00 -0.00000 512.54282 0.000001 11 20.00 -0.00000 515.54560 -0.000000 12 22.00 -0.00000 518.89185 -0.000000 13 24.00 -0.00000 522.59015 0.000001

192

N FI [deg ]

ALFA [deg]

R [mm]

REL. ERROR [%]

14 26.00 -0.00000 526.65006 -0.000000 15 28.00 -0.00000 531.08223 0.000001 16 30.00 -0.00000 535.89838 -0.000001 17 32.00 -0.00000 541.11146 0.000000 18 34.00 -0.00000 546.73562 -0.000001 19 36.00 -0.00000 552.78640 -0.000001 20 38.00 -0.00000 559.28075 -0.000000 21 40.00 -0.00000 566.23717 0.000001 22 42.00 -0.00000 573.67580 0.000000 23 44.00 -0.00000 581.61860 0.000001 24 46.00 -0.00000 590.08943 -0.000001 25 48.00 -0.00000 599.11429 -0.000000 26 50.00 -0.00000 608.72142 0.000001 27 52.00 -0.00000 618.94154 0.000000 28 54.00 -0.00000 629.80809 -0.000000 29 56.00 -0.00000 641.35746 0.000000 30 58.00 -0.00000 653.62926 -0.000000 31 60.00 -0.00000 666.66667 0.000000 32 62.00 -0.00001 680.51674 -0.000000 33 64.00 -0.00001 695.23085 -0.000001 34 66.00 -0.00001 710.86511 -0.000000 35 68.00 -0.00001 727.48087 0.000000 36 70.00 -0.00001 745.14530 0.000000 37 72.00 -0.00001 763.93202 -0.000000 38 74.00 -0.00001 783.92185 -0.000000 39 76.00 -0.00002 805.20361 -0.000001 40 78.00 -0.00000 827.87509 -0.000000 41 80.00 -0.00000 852.04409 -0.000001 42 82.00 -0.00000 877.82972 0.000000 43 84.00 -0.00000 905.36372 -0.000000 44 86.00 -0.00001 934.79219 -0.000000 45 88.00 -0.00001 966.27740 -0.000001 46 90.00 -0.00001 1000.00000 0.000000