Graphics Graphics Lab @ Korea University cgvr.korea.ac.kr 1.2 Notation and Definition 2002. 03. 20 그래픽스 연구실 정병선

Graphics

cgvr.korea.ac.kr Graphics Lab @ Korea University

1.2 Notation and Definition

2002. 03. 20그래픽스 연구실

정병선

cgvr.korea.ac.kr

CGVR

Graphics Lab @ Korea University

Mathematical Definition (1/2)

Type Notation Examples

angle lower-case Greek

scalar lower-case italic

vector or point lower-case bold a,u,vs,h(ρ),hz

matrix capital bold T(t),X,Rx(ρ)

plane π: a vector +

a scalar

π: n·x + d,

π1:n1·x+d1

triangle ∆ 3 points ∆v0v1v2, ∆cba

line segment two points uv,aibj

geometric entity capital italic

242,,,,i

ijk wvutba ,,,,,

AABBOBB BTA ,,

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Mathematical Definition (2/2)

Operator Description

1: ∙ dot product

2: cross product

3: vT transpose of the vector v

4: piecewise vector multiplication

5: ┴ the unary, perp dot product operator

6: | ∙ | determinant of a matrix

7: | ∙ | absolute value of a scalar

8: || ∙ || length (or norm) of argument

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Geometrical Definition

Rendering primitives points, lines, triangles

Model or object a collection of geometric entities may have a higher kind of geometrical

representation. ex) Bezier curves or surfaces, NURBS, subdivision

surfaces, etc.

Scene a collection of models with environment include material descriptions, lighting, viewing

specifications.

Graphics


Appendix A -Some Linear Algebra

2002. 03. 20그래픽스 연구실

정병선

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Euclidean Space (1/6)

Vector Rn : n-dimensional real Euclidean space v : an n-tuple, i.e. an ordered list of real numbers column-major form

1,,0,

1

1

0

niRvwith

v

v

v

vRv i

n

n

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Addition

Multiplication by a scalar

n

nnnn

R

vu

vu

vu

v

v

v

u

u

u

vu

11

11

00

1

1

0

1

1

0

n

n

R

au

au

au

au

1

1

0

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Associativity

Commutativity

Zero identity

Additive inverse

)()( wvuwvu

uvvu

vv 0

0)( vv

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Scalar multiple associativity

Distributive law

Multiplicative identity

)()( buauab

buauuba )(

avauvua )(

uu 1

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Dot product

It’s rules

1

0

n

i iivuvu

vuvu

uvvu

vuavau

wvwuwvu

uifonlyandifuuwithuu

0

)()(

)(

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

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Norm

It’s rules

)(1

0

2

n

i iuuuu

)(

)(

0)0,,0,0(0

inequalitySchwartzCauchyvuvu

inequalitytrianglevuvu

uaau

uu

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Geometrical Interpretation (1/10)

Linearly independent

Span the Euclidean space Rn

tindependenlinearlyareuuvectorsthe

vvvscalarstheifonly

uvuv

n

n

nn

,,,,

0,

0

10

110

1100

nn

n

n

iii

n

RspaceEuclideanthespantosaidareuu

tindependenlinearlyuuvectorstheuvvRv

10

10

1

0

,,

:,,,,,

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Basis of Rn

Dimension of space The largest number of linearly independent vectors in

the space

nn

n

iiin

n

Rofbasisauu

uvvthatsuchvvscalarstheRv

:,,

,,,!,

10

1

010

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Illustration of 3D vector

v = (v0, v1, v2) u0, u1, u2 : bases right-handed system

u0

u1

u2

v

v0

v1

v2

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vector(u) + vector(v)

scalar(a) x vector(w)

u+v

u

vu+v

v

u

w w

aw

-aw

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Illustration of dot product

u

v

20

200

0

cos

ifvu

ifvu

vuvu

vuvu

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Orthonormal basis

Orthogonal basis

Standard basis

ji

jiuu ji ,1

,0

jiuu ji ,1

TTT eee )1,0,0(,)0,1,0(,)0,0,1( 210

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Orthogonal projection

u-wu

w

v

scalart

vectorswvu

tvvvv

vuv

v

vuw

:

:,,

,2

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Cross product

w=uxv

u

v

systemhandedrightaformwvu

vwanduw

vuvuw

,,

sin

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Laws of calculation of cross product

)()()()(

)(

)()(

)()(

)()(

)()()()(

)(

producttriplevectorwvuvwuwvu

producttriplescalar

uvwvwu

wuvvuw

uwvwvu

linearitywvbwuawbvau

itycommutativantiuvvu

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Equation of cross product

Sarrus’s scheme

xyyx

zxxz

yzzy

z

y

x

vuvu

vuvu

vuvu

vu

w

w

w

w

zyxzyx

zyxzyx

zyxzyx

vvvvvv

uuuuuu

eeeeee

+ + + - - -

)()()(

)()()(

xyzzxyyzx

yxzxzyzyx

vuevuevue

vuevuevuevu

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Matrices (1/15)

p x q matrix p rows and q columns

ijqppp

q

q

m

mmm

mmm

mmm

M

1,11,10,1

1,11110

1,01200

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Matrices (2/15)

Identity(unit) matrix matrix-form counterpart of scalar number one square matrix

10000

01000

00010

00001

I

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Matrices (3/15)

Matrix-Matrix addition M + N = [mij] + [nij] = [mij + nij]

(L + M) + N = L + (M + N) M + N = N + M M + 0 = M M – M = 0

Scalar-Matrix multiplication T = aM = [amij]

0M = 0 1M = M a(bM) = (ab)M a0 = 0 (a + b)M = aM + bM a(M + N) = aM + aN

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Matrices (4/15)

Transpose of matrix MT = [mji]

(aM)T = aMT

(M + N)T = MT + NT

(MT)T = M (MN)T = NTMT

Trace of matrix

1

0

)(n

iiimMtr

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Matrices (5/15)

Matrix-Matrix multiplication

1

0 1,,1

1

0 0,,1

1

0 1,,0

1

0 0,,0

1,10,1

1,000

1,10,1

1,000

q

i riip

q

i iip

q

i rii

q

i ii

rqq

r

qpp

q

nmnm

nmnm

nn

nn

mm

mm

MNT

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Matrices (6/15)

Matrix-Vector multiplication

1

0

1

0

1

0 ,1

1

0 ,0

1

0

1,10,1

1,000

ppq

k kkp

q

k kk

pqpp

q

w

w

vm

vm

vm

vm

v

v

mm

mm

Mvw

0w

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Matrices (7/15)

Rules for matrix-matrix multiplication (LM)N = L(MN) (L + M)N = LN + MN MI = IM = M In general, MN ≠ NM

Determinant of matrix 2 x 2 matrix

100111001110

0100 mmmmmm

mmM

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Matrices (8/15)

3 x 3 matrix

zyxzyx mmmmmmmmmM

mmmmmmmmm

mmmmmmmmm

mmm

mmm

mmm

M

)(2,1,0,

211200221001201102

211002201201221100

222120

121110

020100

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Matrices (9/15)

Rules of determinant calculation |M-1| = 1 / |M| |MN| = |M| |N| |aM| = an|M| |MT| = |M| If M = [ami,] or M = [am,j], then |M| = a|M|.

If for i ≠ j, mi, = mj, or m,i = m,j , then |M| = 0.

If for some i, mi, = 0 or m,i = 0, then |M| = 0.

Orientation of basis |bases| > 0 : right-handed system |bases| < 0 : left-handed system

ex) |ex ey ez| = 1 > 0

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Matrices (10/15)

Adjoint matrix subdeterminant

adjoint

1,11,11,110

1,11,11,10,1

1,11,11,10,1

1,01,01,000

nnjnjnn

nijijii

nijijii

njj

Mij

mmmm

mmmm

mmmm

mmmm

d

Mji

jiijij daaMadj )()1(,)(

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Matrices (11/15)

Inverse of matrix must |M| ≠ 0 If MN = I and NM = I, then N = M-1.

Implicit method u = Mv gives v = M-1u Cramer’s rule

1,1,1,1,0,

niii

ii

mmummmd

M

dv

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Matrices (12/15)

Solution by Cramer’s rule for 3 x 3 system

Explicit method Gaussian elimination

Mu = Iv

General case

),,det(

),,det(

),,det(1

umm

mum

mmu

Mv

v

v

v

yx

zx

zy

z

y

x

)(11 MadjM

M

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Matrices (13/15)

Import rules of inverse (M-1)T = (MT)-1

(MN)-1 = N-1M-1

Eigenvalue and Eigenvector Ax = λx (A : square matrix, x : vector, λ : scalar)

x : eigenvector λ : eigenvalue

Theoretical results

orthogonalrseigenvectorealseigenvaluesymmetricrealA

AaAtrn

i i

n

i i

n

i ii

:,:,:)3(

)det()2()()1(1

0

1

0

1

0

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Matrices (14/15)

Orthogonal matrices If MMT = MTM = I, then the sqaure matrix M is orthog

onal. Significant implications

|M| = + 1 M-1 = MT

MT : orthogonal ||Mu|| = ||u|| Mu ┴ Mv iff u ┴ v If M, N are orthogonal , then MN is orthogonal.

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Matrices (15/15)

Change of base current coordinate system to another coordinate syst

em Fw = ( fx fy fz )w = v w = F-1v

If F is orthogonal, F-1 = FT.

v

f

f

f

vFwTz

Ty

Tx

T

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Homogeneous Notation

Point and Vector p = (px, py, pz, pw)T

point : pw = 1

vector : pw = 0

Rotation, scaling, shearing and translation

1000

0

0

0

222120

121110

020100

44 mmm

mmm

mmm

M

1000

100

010

001

z

y

x

t

t

t

T

rotation, scaling, shear matrix translation matrix

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Geometry (1/8)

Two-dimensional line (L)

d

or(t)

td = r(t) - o L

Figure of r(t) = o + td

n

pqL

Figure of n · (p – q) = 0

Explicit form Implicit form

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Geometry (2/8)

Half-plane test ( f(p) = n · p + c, c = -q · n )

Signed distance

.int0)(.3

.int0)(.2

.0)(.1

nqpotheaslinetheofsidesametheonliesppf

nqpotheaslinetheofsidesametheonliesppf

Lppf

)()(1

)()(

pfpfnnn

pfpf

s

s

Lp

n

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Geometry (3/8)

Three-dimensional line same as two-dimensional line except for 3D Distance p to r(t)

p

od

w

||(p – o) – w||

r(t)

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Geometry (4/8)

Planes (π)

Explicit form Implicit form

du

dv

o

p(u,v) = o + udu+ vdv

udu

vdv

p

q

n

Figure of p(u,v) = o + udu+ vdv Figure of n · ( p – q ) = 0

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Geometry (5/8)

Half-plane test ( f(p) = n · p + d, d = -n · q )

Signed distance obtained by exchanging the two-dimensional parts of the eq

uation for their three-dimensional counterparts for the plane. fs(0) = d, d : the shortest signed distance from the origin to th

e plane.

.int0)(.3

.int0)(.2

.0)(.1

nqpotheasplanetheofsidesametheonliesppf

nqpotheasplanetheofsidesametheonliesppf

ppf

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Geometry (6/8)

Convex hull the smallest set such that the straight line between

any two points in the set is totally included in the set as well.

rubber band

convex hull

release

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Geometry (7/8)

Area calculation

)()(2

1)( rqrppqrArea

||v||sinΦ

Φ

v

u

sinvuvu

p

q r

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Geometry (8/8)

Volume calculation

u x v

w

v

u

Φ

wvuwvuwvuVolume )(),,(

Graphics


Chapter 3 - Transforms

2002. 03. 20그래픽스 연구실

정병선

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Basic Transforms (1/12)

Notation Name

T(t) translation matrix

Rx(ρ) rotation matrix

R rotation matrix

S(s) scaling matrix

Hij(s) shear matrix

E(h,p,r) Euler transform

Po(s) orthographic projection

Pp(s) perspective projection

slerp(q,r,t) slerp transform

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Translation matrix

example

)()(,

1000

100

010

001

),,()( 1 tTtTt

t

t

tttTtTz

y

x

zyx

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Rotation matrix

)(,

1000

0100

00cossin

00sincos

)(

)(,

1000

0cos0sin

0010

0sin0cos

)(

)(,

1000

0cossin0

0sincos0

0001

)(

1

1

1

zz

yy

xx

RR

RR

RR

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example

)()()( pTRpTX z

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Scaling matrix

example scaling in a certain direction

1000

01

00

001

0

0001

)(,

1000

000

000

000

)( 1

z

y

x

z

y

x

s

s

s

sSs

s

s

sS

T

zyx

zyx

FsFSX

vectorsorientedrightlorthonormathefff

fffF

)(

:,,

1000

0

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Shearing matrix

6 basic shearing matrices

)()(,

1000

0100

0010

001

)( 1 sHsH

s

sH xzxzxz

)(),(),(),(),(),( sHsHsHsHsHsH zyzxyzyxxzxy

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example

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Concatenation of transforms example

Rotating a unit-square π/6 radians by shearing three times

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Order dependency multiplication of matrices is not commutative. C = TRS example

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Rigid-body transform the shape of the object is not affected by transform. so, this transform consists of translations and

rotations. matrix

)()())((

1000

)(

1111

222120

121110

020100

tTRtTRRtTX

trrr

trrr

trrr

RtTX

T

x

x

x

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Normal transform The surface normal must be transformed by the

transpose of the inverse of the matrix used to transform geometry.

example

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Computation of inverses inversion of parameters

M = T(t)R(Φ) → M-1 = R(- Φ)T(-t)

orthogonal matrix M-1 = MT

nothing in particular Cramer’s rule Gaussian elimination

Documents

Graphics Graphics Lab @ Korea University cgvr.korea.ac.kr 1.2 Notation and Definition 2002. 03. 20 그래픽스 연구실 정병선