Farouk i

8/9/2019 Farouk i

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Spatial Pythagorean hodographs,quaternions, and rotations in R

3 and R4

— interspersed with historical vignettes —

“for your erudition and entertainment”

Rida T. Farouki

Department of Mechanical & Aeronautical Engineering,University of California, Davis

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— synopsis —

• motivation – real-time interpolators for CNC machines

• background – fundamentals of the quaternion algebra

• interlude #1 – unit quaternions and rotations inR

3

• basic theory – complex number & quaternion formulations

of planar & spatial Pythagorean-hodograph curves

• interlude #2 – unit quaternions and rotations in R4

• applications – first-order PH quintic Hermite interpolants

& computation of rotation-minimizing frames

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3-axis “open architecture” CNC mill

• MHO Series 18 Compact Mill

• 18” × 18” × 12” work volume

• Yaskawa brushless DC motors

• zero-backlash precision ball screws

• linear encoders, ± 0.001” accuracy

• MDSI OpenCNC control software

• custom real-time interpolators

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“conventional” vs. “high-speed” machining

• machining times are major determinant of product cost

• usual: feedrate ≤ 100 in/min, spindle speed ≤ 6, 000 rpm

• HSM: feedrate ≤ 1, 200 in/min, spindle speed ≤ 50, 000 rpm

• in HSM inertial effects may dominate cutting forces, friction, etc.(especially for intricate curved tool paths)

• most CNC machines significantly under-perform in practice

– control software, not hardware, is the limiting factor

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real-time CNC interpolators

• computer numerical control (CNC) machine has digital control system

• in each sampling interval (∆t ∼ 10−3 sec) of servo system, compareactual position (measured by encoders on each machine axis) withreference position (computed by real-time interpolator algorithm)

• real-time CNC interpolator algorithm — given parametric curve r(ξ)and speed (feedrate) function v, compute reference-point parametervalues ξ1, ξ2, . . . in real time:

ξk

0

|r(ξ)| dξ

v = k∆t , k = 1, 2, . . .

• Pythagorean-hodograph (PH) curves — analytic reduction of“interpolation integral” =⇒ accurate & efficient real-time interpolator

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real-time CNC interpolators

for Pythagorean-hodograph (PH) curves

0 4 8 12 16

4

8

12

p0

p1

p4

p5

25 segments

ε = 0.0105

50 segments

ε = 0.0025

100 segments

ε = 0.0006

Left: analytic tool path description (quintic PH curve). Right: approximationof path to various prescribed tolerances using piecewise-linear G codes.

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comparative feedrate performance: 100 & 800 ipm

0 1 20

200

400

600

800

time (seconds)

f e e d r a t e ( i p m )

G codes

800 ipm

0 1 2 3 4 5 6 7 80

50

100

150

200

f e e d r a t e ( i p m )

G codes

100 ipm

0 1 2

time (seconds)

PH curve

800 ipm

PH curve

100 ipm

0 1 2 3 4 5 6 7 8

Both G code and PH curve interpolators give excellent performance at 100 ipm — the

“staircase” nature of the x and y feedrate components (red and green) for the G codes

indicate faithful reproduction of the piecewise-linear path, while the PH curve yields

smooth variations. At 800 ipm, the PH curve interpolator gives impeccable performance,

but the performance of the G code interpolator is severely degraded by “aliasing” between

the finite sampling frequency and discrete nature of the path description.

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optimal section-plane orientation

for contour machining of free-form surfaces

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parabolic lines on free-form surfaces

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Gauss map computation for free-form surfaces

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medial axis transform for complement of Gauss map

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fundamentals of quaternion algebra

quaternions are four-dimensional numbers of the form

A = a + ax i + ay j + az k and B = b + bx i + by j + bz k

that obey the sum and (non-commutative) product rules

A + B = (a + b) + (ax + bx) i + (ay + by) j + (az + bz)k

A B = (ab − axbx − ayby − azbz)

+ (abx + bax + aybz − azby) i

+ (aby + bay + azbx − axbz) j

+ (abz + baz + axby − aybx)k

basis elements 1, i, j, k satisfy i2 = j2 = k2 = i j k = −1

equivalently, i j = − j i = k , j k = −k j = i , k i = − i k = j

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scalar-vector form of quaternions

set A = (a, a) and B = (b,b) — a, b and a, b are scalar and vector parts

(a, b and a,b also called the real and imaginary parts of A, B)

A + B = ( a + b , a + b )

A B = ( ab − a · b , ab + ba + a × b)

(historical note: Hamilton’s quaternions preceded, but were eventuallysupplanted by, the 3-dimensional vector analysis of Gibbs and Heaviside)

A∗ = (a, −a) is the conjugate of A

modulus : |A |2 = A∗A = AA∗ = a2 + |a|2

note that | A B| = |A||B| and (A B)∗ = B∗A∗

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unit quaternions & spatial rotations

any unit quaternion has the form U = (cos

1

2θ, sin

1

2θ n)

describes a spatial rotation by angle θ about unit vector n

for any vector v the quaternion product

v = U v U ∗

yields the vector v corresponding to a rotation of v by θ about n

here v is short-hand for a “pure vector” quaternion V = (0,v)

unit quaternions U form a (non-commutative) group under multiplication

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concatenation of spatial rotations

rotate θ1 about n1 then θ2 about n2 → equivalent rotation θ about n

θ = ± 2 cos−1(cos 12θ1 cos 1

2θ2 − n1 · n2 sin 12θ1 sin 1

2θ2)

n = ±sin 1

2θ1 cos 12θ2 n1 + cos 1

2θ1 sin 12θ2 n2 − sin 1

2θ1 sin 12θ2 n1 × n2

1 − (cos

12θ1 cos

12θ2 − n1 · n2 sin

12θ1 sin

12θ2)

2

sign ambiguity: equivalence of − θ about −n and θ about n

formulae discovered by Olinde Rodrigues (1794-1851)

set U 1 = (cos 12θ1, sin 1

2θ1n1) and U 2 = (cos 12θ2, sin 1

2θ2n2)

U = U 2 U 1 = (cos 12θ, sin 1

2θn) defines angle, axis of compound rotation

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spatial rotations do not commute

x y

z

x y

z

α

β

α

β

blue vector is obtained from red vector by the concatenation of two spatialrotations — left: Ry(α) Rz(β ), right: Rz(β ) Ry(α) — the end results differ

define U 1 = (cos 12α, sin 1

2α j), U 2 = (cos 12β, sin 1

2β k) — U 1 U 2 = U 2 U 1

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the “troubled origins” of vector analysis

The algebraically real part may receive . . . all values contained on the

one scale of progression of number from negative to positive infinity; weshall call it therefore the scalar part , or simply the scalar . On the otherhand, the algebraically imaginary part, being constructed geometricallyby a straight line or radius, which has, in general, for each determinedquaternion, a determined length and determined direction in space, maybe called the vector part , or simply the vector . . .

William Rowan Hamilton, Philosophical Magazine (1846)

A school of “quaternionists” developed, which was led after Hamilton’sdeath by Peter Tait of Edinburgh and Benjamin Pierce of Harvard. Taitwrote eight books on the quaternions, emphasizing their applications to

physics. When Gibbs invented the modern notation for the dot and crossproduct, Tait condemned it as a “hermaphrodite monstrosity.” A war ofpolemics ensued, with luminaries such as Kelvin and Heaviside writingdevastating invective against quaternions. Ultimately the quaternions lost,and acquired a taint of disgrace from which they never fully recovered.

John C. Baez, The Octonions (2002)

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the sad demise of quaternions

E. T. Bell, Men of Mathematics , Hamilton = “An Irish Tragedy”

Hamilton’s Lectures on Quaternions (1853) “would take any

man a twelve-month to read, and near a lifetime to digest . . .”

– Sir John Herschel, discoverer of the planet Uranus

Hamilton’s vision of quaternions as the “universal language”

of mathematical and physical sciences was never realized —

this role is now occupied by vector analysis, distilled from thequaternion algebra by the physicists James Clerk Maxwell

(1831-1879) and Josiah Willard Gibbs (1839-1903), and the

engineer Oliver Heaviside (1850-1925)

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families of spatial rotations

find U = (cos

1

2θ, sin

1

2θ n) that rotates i = (1, 0, 0) → v = (λ,µ,ν )

n2x(1 − cos θ) + cos θ = λ ,

nxny(1 − cos θ) + nz sin θ = µ ,

nznx(1 − cos θ) − ny sin θ = ν .

nx =±

cos2 1

2α − cos2 1

2θ

sin 12

θ,

ny =

±µ cos2 12

α − cos2 12

θ − ν cos 12

θ

(1 + λ) sin 12θ ,

nz =± ν

cos2 1

2α − cos2 1

2θ + µ cos 1

2θ

(1 + λ) sin 12

θ.

general solution, where α = cos−1 λ and α ≤ θ ≤ 2π − α

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Parameterizes family of spatial rotations mapping unit vectors i → v byspecifying rotation axis n as a function of rotation angle θ, over restricteddomain θ ∈ [ α, 2π − α ] where α is angle between i and v.

Define unit vectors e⊥, e0 orthogonal to and in common plane of i and v

e⊥ =i × v

| i × v |and e0 =

i + v

| i + v |

Rotation axis lies in plane spanned by these vectors, may be written as

n(θ) =sin 1

2α cos 12θ e⊥ ±

cos2 1

2α − cos2 12θ e0

cos 12α sin 1

2θ.

for any θ ∈ (α, 2π − α) there are two axes n — in the plane of e⊥, e0 withequal inclinations to e⊥ — about which a rotation by angle θ maps i → v

◦ when θ = α or 2π − α, we have n = e⊥ or −e⊥, and rotation isalong great circle between i and v;

◦ when θ = π, we have n = ± e0, so i executes either a clockwise

or anti–clockwise half-rotation about e0 onto v;

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(a) (b)

(c) (d)

e0

e⊥

i

v

i

e⊥ v

n

n

Spatial rotations of unit vectors i → v. (a) Vectors e⊥ (orthogonal to i, v)and e0 (bisector of i,v) — the plane Π of e⊥ and e0 is orthogonal to that ofi and v. (b) For any rotation angle θ ∈ (α, 2π − α), where α = cos−1(i · v),there are two possible rotations, with axes n inclined equally to e⊥ in theplane Π. (c) Sampling of the family of spatial rotations i → v, shown as

loci on the unit sphere. (d) Axes n for these rotations, lying in the plane Π.

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Pythagorean-hodograph (PH) curves

r(ξ) = PH curve ⇐⇒ coordinate components of r

(ξ)comprise a “Pythagorean n-tuple of polynomials” in R

n

PH curves exhibit special algebraic structures in their hodographs

• rational offset curves rd(ξ) = r(ξ) + dn(ξ)

• polynomial arc-length function s(ξ) =

ξ

0

|r(ξ)| dξ

• closed-form evaluation of energy integral E =

1

0

κ2 ds

• real-time CNC interpolators, rotation-minimizing frames

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Pythagorean triples of polynomials

x2(t) + y2(t) = σ2(t) ⇐⇒

x(t) = u2(t) − v2(t)y(t) = 2 u(t)v(t)

σ(t) = u2(t) + v2(t)

key features of planar PH curves

◦ planar PH cubics come from unique curve — Tschirnhausen’s cubic

◦ planar PH quintics can interpolate arbitrary Hermite data

solve nested quadratic equations — always four distinct solutions

◦ extend to planar C 2 PH quintic splines — “tridiagonal” systemof 2N ±1 quadratic equations in N complex unknowns

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Plimpton 322

origin — Larsa (Tell Senkereh) in Mesopotamia ∼ 1820–1762 BC

discovered in 1920s — bought in market by dealer Edgar A. Banks — soldto collector George A. Plimpton for $10 — donated to Columbia University

deciphered in 1945 by Otto Neugebauer and Abraham Sachs — but

significance, meaning, or “purpose” still the subject of great controversy

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sketch of Plimpton 322 by Eleanor Robson

fifteen rows of sexagecimal numbers in four columns

3, 31, 49 → 3 × (60)2 + 31 × 60 + 49

1; 48, 54, 1, 40 → 1 +48

60+

54

(60)2+

1

(60)3+

40

(60)4

first three columns generated by integers p, q through formulae

f =

p2 + q2

2 pq

2

, s = p2− q2 , d = p2 + q2

with 1 < q < 60, q < p, p/q steadily decreasing

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f = [ ( p2 + q2)/2 pq ]2 s = p2− q2 d = p2 + q2 #

[1;59,0,]15 1,59 2,49 1

[1;56,56,]58,14,50,6,15 56,7 1,20,25 2

[1;55,7,]41,15,33,45 1,16,41 1,50,49 3

[1;]5[3,1]0,29,32,52,16 3,31,49 5,9,1 4

[1;]48,54,1,40 1,5 1,37 5

[1;]47,6,41,40 5,19 8,1 6

[1;]43,11,56,28,26,40 38,11 59,1 7

[1;]41,33,59,3,45 13,19 20,49 8

[1;]38,33,36,36 8,1 12,49 9

1;35,10,2,28,27,24,26,40 1,22,41 2,16,1 10

1;33,45 45,0 1,15,0 111;29,21,54,2,15 27,59 48,49 12

[1;]27,0,3,45 2,41 4,49 13

1;25,48,51,35,6,40 29,31 53,49 14

[1;]23,13,46,40 56 1,46 15

p q

12 5

1,4 27

1,15 32

2,5 549 4

20 9

54 25

32 15

25 12

1,2 1 40

1,0 3048 25

15 8

50 27

9 5

Pythagorean triples of integers

s2 + l2 = d2 ⇐⇒

s = p2 − q2

l = 2 pqd = p2 + q2

l = 2 p q

s

=

p 2

–

q 2

d = p 2 + q 2

θ

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significance of Plimpton 322

R. C. Buck (1980), Sherlock Holmes in Babylon , Amer. Math. Monthly 87, 335-345

• investigate in isolation as a “mathematical detective story”

• an exercise in number theory (s, l , d) = ( p2− q2, 2 pq, p2 + q2) ?

• construction of a trigonometric table — sec2 θ = [( p2 + q2)/2 pq] 2 ?

Eleanor Robson (2001), Neither Sherlock Holmes nor Babylon —

A Reassessment of Plimpton 322 , Historia Mathematica 28, 167-206

• studied mathematics, then Akkadian and Sumerian at Oxford

• linguistic, cultural, historical context critical to a proper interpretation

• number theory & trigonometry interpretations improbable — more likely

a set of “cut–and–paste geometry” exercises for the training of scribes

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“cut-and-paste geometry” problem

find regular reciprocals x,1

xsatisfying x =

1

x+ h for integer h

x = 1 / x + h

1

/ x

h / 2

1 / x + h / 2

1

/ x

+

h

/ 2

“cut-and-paste geometry” problem : 1 =

1

x+

h

2

2

−

h

2

2

writing x = pq

, 1x

= q p

gives 1x

+ h2

= 12

pq

+ q p

, h

2= 1

2

pq

− q p

scaling by 2 pq yields d = p2 + q2, s = p2 − q2

f represents (unscaled) area of large square

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complex model for planar PH curves

w w2

w(t) = u(t) + i v(t) maps to r(t) = w2(t) = u2(t) − v2(t) + i 2 u(t)v(t)

rotation invariance of planar PH form: rotate by θ, r(t) → r(t)

then r(t) = w2(t) where w(t) = u(t) + i v(t) = exp(i 12θ)w(t)

in other words,

u(t)v(t)

=

cos 1

2θ − sin 12θ

sin 12θ cos 1

2θ

u(t)v(t)

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Bezier curve

P. de Casteljau (Citroen) – P. Bezier (Renault)

convex hull

subdivision

variation diminishing

numerical stability

p0

p1

p2

p3

p4

p5

r(t) = ∑k=0

n

pk nk

(1–t)n–ktk

Bernstein basis on [0,1] :

[ (1–t)+t]n = (1–t)n + n(1–t)n–1t + ... + tn

Farouki’s Tschirnhausen’s (1690) cubic

PH cubic iff L22

= L1L3 & θ1 = θ2 (1990)

unique curve !

L1

L2

L3

θ1

θ2

caustic forreflection

by parabola

trisectrix of Catalan

l’Hospital’s cubic

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Ehrenfried Walther von Tschirnhaus 1651–1708

◦ contemporary of Huygens, Leibniz, and Newton

◦ visited London and Paris after studying in Leiden

◦ investigated burning mirrors in Milan and Rome

• Tschirnhaus transform “A method for eliminating all intermediateterms from a given equation” — Acta Eruditorum, May 1683

• empirical & analytical investigations of caustics by reflection

• Tschirnhausen’s cubic = unique cubic Pythagorean-hodograph curve

• developed manufacture of hard–fired porcelain in Dresden

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Tschirnhaus transform of cubic equation

t

3

+ a2t

2

+ a1t + a0 = 0

Descartes: t → t − 13a2 eliminates t2 term

Tschirnhaus considers cubics of the form t3 = q t + r

and defines transformation t → τ by t =2qa − 3r + 3aτ

q − 3a2 − 3τ ,

where a is a root of the quadratic 3q a2 − 9r a + q2 = 0

simplification gives τ 3 =(27r2 − 4q3)(2q2 − 9ra)

27q2

Bing–Jerrard “reduced form” of quintic: t

5

= q t + r

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caustics for reflection by a sphere and a parabola

left: epicycloid right: Tschirnhausen’s cubic

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Tschirnhausen’s cubic = negative pedal of parabola with respect to focus

Tschirnhausen’s cubic = trisectrix of Catalan — ∠P F Q =1

3∠OF Q

O F

P

Q

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Pythagorean quartuples of polynomials

x2(t) + y2(t) + z2(t) = σ2(t) ⇐⇒

x

(t) = u2

(t) + v2

(t) − p2

(t) − q2

(t)y(t) = 2 [ u(t)q(t) + v(t) p(t) ]

z(t) = 2 [ v(t)q(t) − u(t) p(t) ]

σ(t) = u2(t) + v2(t) + p2(t) + q2(t)

R. Dietz, J. Hoschek, and B. Juttler, An algebraic approach to curves and surfaces on the sphereand on other quadrics, Computer Aided Geometric Design 10, 211–229 (1993)

H. I. Choi, D. S. Lee, and H. P. Moon, Clifford algebra, spin representation, and rational

parameterization of curves and surfaces, Advances in Computational Mathematics 17, 5-48 (2002)

choose quaternion polynomial A(t) = u(t) + v(t) i + p(t) j + q(t)k

→ spatial Pythagorean hodograph r(t) = (x(t), y(t), z(t)) = A(t) iA∗(t)

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quaternion model for spatial PH curves

quaternion polynomial A(t) = u(t) + v(t) i + p(t) j + q(t)k

maps to r(t) = A(t) iA∗(t) = [ u2(t) + v2(t) − p2(t) − q2(t) ] i

+ 2 [ u(t)q(t) + v(t) p(t) ] j + 2 [ v(t)q(t) − u(t) p(t) ]k

rotation invariance of spatial PH form: rotate by θ about n = (nx, ny, nz)

define U = (cos 12θ, sin 1

2θ n) — then r(t) → r(t) = A(t) i A∗(t)

where A(t) = U A(t) (can interpret as rotation in R4)

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matrix form of A(t) = U A(t)

u

v

˜ p

q

=

cos 12θ −nx sin 1

2θ −ny sin 12θ −nz sin 1

2θ

nx sin 12θ cos 1

2θ −nz sin 12θ ny sin 1

2θ

ny sin 12θ nz sin 1

2θ cos 12θ −nx sin 1

2θ

nz sin 12θ −ny sin 1

2θ nx sin 12θ cos 1

2θ

u

v

p

q

matrix ∈ SO(4)

in general, points have non-closed orbits under rotations in R4

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“strange events” in R4 defy geometric intuition !

◦ an elastic sphere can be turned inside out without tearing the material !

◦ a prisoner may escape from a locked room without penetrating its walls !

◦ rigid motions change “left-handed objects” into “right-handed objects” !

◦ a knot in a length of string can be untied without ever moving its ends !

early 20th century: can existence of a fourth dimension, imperceptible tohuman senses, explain mysterious psychic and paranormal phenomena?

H. P. Manning, The Fourth Dimension Simply Explained (1910)& Geometry of Four Dimensions (1914), Dover (reprint), New York.

→ strange phenomena arise from “extra maneuvering freedom” inR

4

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elementary geometry of four dimensions

lines, planes, and hyperplanes ofR

4

are the sets of points linearlydependent upon two, three, and four points of R4 in “general position”

alternatively, lines, planes, and hyperplanes are point sets satisfyingthree, two, and one linear equations in the Cartesian coordinates of R4

hyperplane = a copy of familiar Euclidean space R3 — separatesR

4 into two disjoint regions (as with a plane in R3, and a line in R2)

generic incidence relations for R4 :

◦ two hyperplanes intersect in a plane

◦ three hyperplanes intersect in a line

◦ four hyperplanes intersect in a point=⇒ two planes intersect in a point

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“absolutely orthogonal” planes in R4

two planes Π1, Π2 ∈ R4 with intersection point p are absolutely orthogonalif every line through p on Π1 is orthogonal to every line through p on Π2

pairs of “absolutely orthogonal” planes are a strictly four-dimensionalphenomenon — they have no analog in R3

through each point p of a given plane Π1 ∈ R4, there is a unique planeΠ2 ∈ R4 that is absolutely orthogonal

pairs of absolutely orthogonal planes in R4 play an important role in

characterizing four-dimensional rotations

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characterization of rotations in R2, R3, R4

R2

: (x + i y) → eiθ(x + i y) — one parameter, rotation angle θ

R3 : v → U v U ∗ where U = (cos 1

2θ, sin 12θ n)

— three parameters, rotation axis n and angle θ

R

4

: V → U 1V U ∗

2 — two unit quaternions U 1, U 2 ⇒ six parameters

stationary set of rotation in Rn = set of points that do not move

simple rotation in Rn — the stationary set is of dimension n − 2

⇒ in R2 and R3, every rotation is simple

simple rotation in R4 — stationary set is a plane through the origin,

and unique absolutely orthogonal plane rotates on itself

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a new possibility in R4 — double rotations

if Π1, Π2 ∈ R4

are absolutely orthogonal planes through the origin, eachmay rotate upon itself about the other, and these rotations commute —i.e., the outcome is independent of their order

the stationary set of such a double rotation is the single common pointof Π

1, Π

2— i.e., the origin

of the six parameters describing a general rotation in R4, four define theabsolutely orthogonal planes Π1, Π2 and two specify the rotation anglesθ1, θ2 associated with them

under a continuous double rotation — with angular speeds ω1, ω2

associated with the absolutely orthogonal planes Π1, Π2 — points in R4

have closed orbits if and only if the ratio ω2/ω1 is a rational number

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spatial PH quintic Hermite interpolants

spatial PH quintic interpolating end points pi, pf & derivatives di, df

r(t) = A(t) iA∗(t)

where A(t) = A0(1 − t)2 + A12(1 − t)t + A2t2

three equations in three quaternion unknowns A0, A1, A2

r(0) = A0 iA∗0 = di and r(1) = A2 iA∗

2 = df

1

0

A(t) iA∗(t) dt = 1

5

A0 iA∗

0 + 1

10

(A0 iA∗

1 + A1 iA∗

0)

+ 130(A0 iA∗

2 + 4 A1 iA∗

1 + A2 iA∗

0)

+ 110(A1 iA∗

2 + A2 iA∗

1) + 15 A2 iA∗

2 = pf − pi

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solution of fundamental equation

given vector c = |c|(λ,µ,ν ) find quaternion A such that

A iA∗ = c

one–parameter family of solutions

A(φ) =

12(1 + λ)|c|

− sin φ + cos φ i

+µ cos φ + ν sin φ

1 + λj +

ν cos φ − µ sin φ

1 + λk

in R3 there is a continuous family of rotations

mapping the vector i into a given vector (λ,µ,ν )

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construction of Hermite interpolants

derivative conditions have form of fundamental equation

— can be solved directly for A0 and A2

end-point condition can then be cast in fundamental form as

(3A0 + 4A1 + 3A2) i (3A0 + 4A1 + 3A2)∗

= 120(pf − pi) − 15(di + df ) + 5(A0 iA∗

2 + A2 iA∗

0)

— solve for A1, since A0 and A2 known

solution contains three free parameters φ0, φ1, φ2but shape of interpolants depends only on their differences

=⇒ ∃ two-parameter family of spatial PH quintic interpolants

to given Hermite data pi, di and pf , df

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spatial PH quintic Hermite interpolants

pi = (0, 0, 0) and pf = (1, 1, 1) for both curves

di = (−0.8, 0.3, 1.2) and df = (0.5, −1.3, −1.0) for curve on left,

di = (0.4, −1.5, −1.2) and df = (−1.2, −0.6, −1.2) for curve on right

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open problem: find “optimal” φ0, φ2 values

shape of interpolants depends strongly on free parameters

• minimize a shape-measure integral, e.g., E =

κ2 ds

(but highly non-linear in the free parameters)

• impose additional conditions (restrict solution space) — e.g., helicity condition κ/τ = constant

• study geometry of quaternion curve A(t) — need better insight on geometry of quaternion space H

• extension to spatial C 2 PH quintic splines

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defects of Frenet frame (t,n,b) on space curve

• does not depend rationally on curve parameter

• suffers sudden “flip” at inflection points, where κ = 0

• exhibits “unnecessary rotation” in curve normal plane

dt

ds= d × t ,

dn

ds= d × n ,

db

ds= d × b ,

Darboux vector d = κb + τ t Frenet frame rotation rate

component τ t describes instantaneous rotation in normal plane

(unnecessary for a “smoothly varying” orthonormal frame)

causes problems for animation, path planning, swept surfaces, etc.

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rotation-minimizing frame on spatial PH curves

new basis in normal plane

e2

e3

=

cos θ sin θ

− sin θ cos θ

n

b

where θ = −

τ ds : cancels “unnecessary rotation” in normal plane

options for construction of RMF (t, e2, e3) on spatial PH quintics:

• exact analytic form — involves rational function integration,logarithmic dependence on curve parameter

• rational approximation — use Pade (rational Hermite) approach:simple algorithm & rapid convergence

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comparison of Frenet & rotation-minimizing frames

spatial PH quintic Frenet frame rotation–minimizing frame

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rotation rates — RMF vs Frenet frame

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

1

2

3

t

r a t e

o f r o t a t i o n

rotation minimizingFrenet frame

compared with the rotation-minimizing frame (t, e2, e3), the Frenet frame(t,n,b) exhibits a lot of “unnecessary” rotation (in the curve normal plane)

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closure

• Pythagorean-hodograph curves ideally suited to CNC machining

• PH curve advantages: rational offset curves, analytic real-timeinterpolators, exact rotation-minimizing frame, bending energy, etc.

• complex number and quaternion models are “natural” formulationsfor planar and spatial PH curves — rotation invariance, simplifiedconstruction algorithms, etc.

• spatial PH curve open problems — extra degrees of freedom inHermite interpolation, C 2 spline formulation, better “geometrical”insight into quaternion representation

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man’s limited insight

Superior beings, when of late they saw

A mortal man unfold all nature’s law,

Admired such wisdom in an earthly shape,

And showed a Newton as we show an ape.

Could he, whose rules the rapid comet bind,

Describe or fix one movement of his mind?Who saw its fires here rise, and there descend,

Explain his own beginning, or his end?

Alas, what wonder! Man’s superior part

Unchecked may rise, and climb from art to art:But when his own great work has but begun,

What reason weaves, by passion is undone.

Alexander Pope (1688-1744), Essay on Man

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Boolean algebra of poets & fools

Sir, I admit your general rule,

That every poet is a fool.

But you yourself may serve to show it,

That every fool is not a poet!

Alexander Pope (1688-1744)

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