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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)