Dorst L., Fontijne D., Mann S. Geometric Algebra for Computer Science. An Object Oriented Approach to Geometry
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604 COMMON EQUATIONS APPENDIX D
Table D.1: Notation of products, operators, and possible definitions. For alternative definitions, disambiguation of meet
and join, and other details, we refer to the chapters indicated. In the formulas, I
n
is the unit pseudoscalar for the total
space, and a and b are the grades of the blades A and B. All definitions except
meet and join can be extended from blades’
general multivectors. The
C++ column presents the operators and function names used in the programming examples in
Part I and Part II.
Product or Operation Notation Definition Reference C++
geometric product AB fundamental Chapter 6 ∗
outer product A ∧ B AB
a+b
Chapter 2 ˆ
scalar product A ∗ B AB
0
Chapter 3 %
left contraction AB AB
b−a
Chapter 3 <<
right contraction AB AB
a−b
Chapter 3 >>
commutator product A ×B
2
A B
2
a
Chapter 8 N.A.
cross product (R
3,0
only) a × b (a ∧ b)I
−1
3
Chapter 3 N.A.
meet M A ∩ B (BJ
−1
)A Chapter5,21 meet(A, B)
join J A ∪ B A ∧ (M
−1
B) Chapter5,21 join(A, B)
extraction of grade i A
i
Chapter 2 takeGrade(A, 1 << i)
reverse
A
(−1)
a(a−1)/2
A Chapter 2 reverse(A)
grade involution
A
(−1)
a
A Chapter 2 gradeInvolution(A)
inverse A
−1
A/(A
A)
Chapter3,6 inverse(A)
dual A
∗
A/I
n
Chapter3,6 dual(A)
undual A
−∗
AI
n
Chapter3,6 undual(A)
squared norm A
2
A
A Chapter 3 norm r2(A)
Table D.2: Sign change per grade for reversion, grade involution, and Clifford conjugation.
Operation Notation Multiplier for grade x Pattern
reversion
X
(−1)
x(x−1)/2
++−−++−−
grade involution
X
(−1)
x
+ − + − + − + −
Clifford conjugation X (−1)
x(x+1)/2
+ −−++−−+
APPENDIX D COMMON EQUATIONS 605
Table D.3: Linear transformation f applied to common products.
Product Linear transformation Reference
outer product f[A ∧ B] = f[A] ∧ f[B] (4.3)
scalar product f[A ∗ B] = A ∗ B (4.9)
left contraction f[AB] = f
−1
[A] f[B] (4.13)
dual f
∗
[A
∗
] = det(f)f
−1
[A
∗
] (4.14)
cross product f[a × b] = det(f)f
−1
[a × b] (4.15)
pseudoscalar f[I
n
] = det(f) I
n
(4.7)
meet f[A ∩ B] = f[A] ∩ f[B] (or with det(f)) Section 5.7
join f[A ∪ B] = f[A] ∪ f[B] (or with det(f)) Section 5.7
Table D.4: Basic equations in geometric algebra.
Operation Formula Reference
symmetry of outer product a ∧ B =
1
2
(aB+
Ba) (6.10)
symmetry of outer product B ∧ a =
1
2
(Ba+ a
B) (6.11)
symmetryofleftcontraction aB =
1
2
(aB−
Ba) (6.12)
symmetry of right contraction Ba =
1
2
(Ba− a
B) (6.13)
contractionvectoronblade a(b ∧ C ) = (a · b) ∧ C − b ∧ (aC) from (3.10)
contraction vector on versor a(bC) = (a · b) C − b (aC) (C.5)
contractionvectoronblade x(a
1
∧···∧a
k
) =
k
i=1
(−1)
i−1
a
1
∧···( xa
i
) ∧···a
k
(3.16)
reciprocal frame b
i
= (−1)
i−1
(b
1
∧···
˘
b
i
∧···∧b
n
)I
−1
n
(3.31)
dual X
∗
= XI
−1
n
Section 3.5.3
undual in R
n
(and R
n+1,1
) X
−∗
= (−1)
n(n−1)/2
X
∗
Section 3.5.3
duality relationship (A ∧ B)
∗
= A(B
∗
) (3.24)
duality relationship (AB)
∗
= A ∧ (B
∗
), for A ⊆ I (3.24)
projection of X onto B X → (XB)B
−1
= (XB)/B Section 3.6, 6.4.2
rejection of vector x by B x → (x ∧ B)B
−1
= (x ∧ B)/B Section 3.6, 6.4.2
reflection of direct X in dual D X → (−1)
xd
DXD Table 7.1
applying an even versor V X → V X V
−1
(7.19)
applying an odd versor V X → V
X V
−1
(7.20)
inverseofaversorV V
−1
=
V/(V
V ) (21.1)
606 COMMON EQUATIONS APPENDIX D
Table D.5: Exponential, cosine, and sine of blades. See Section 7.4.
Taylor series Special cases
exp A = 1 +
A
1!
+
A
2
2!
+ ··· exp A =
⎧
⎪
⎨
⎪
⎩
cos α + A
sin α
α
if A
2
= −α
2
1 + A if A
2
= 0
cosh α + A
sinh α
α
if A
2
= α
2
sin A = A −
A
3
3!
+
A
5
5!
−··· sin A =
⎧
⎪
⎨
⎪
⎩
A
sin α
α
if A
2
= α
2
A if A
2
= 0
A
sinh α
α
if A
2
= −α
2
cos A = 1 −
A
2
2!
+
A
4
4!
−··· cos A =
⎧
⎪
⎨
⎪
⎩
cos α if A
2
= α
2
1ifA
2
= 0
cosh α if A
2
= −α
2
Table D.6: Working with rotors.
Requirement Formula Reference
log of Euclidean rotor R log(R) =
R
2
/
R
2
atan2(R
2
,R
0
) (10.14)
shortest rotor from unit a to unit b R = (1 + ba)
/
2 |1 + a · b| (10.13)
rotor/quaternion relationship Section 7.3.5
matrix/rotor conversion Section 7.10.4
Table D.7: Convenient equations from the homogeneous model in Chapter 11.
Requirement Formula Reference
parameters of all blades flats, directions Table 11.3
intersection and spanning Pl
¨
ucker formulas Table 12.1
rigid body motions transformation matrices Table 12.2
translation direct blade X → X + t ∧ (e
−1
0
X) (11.13)
translation dual blade X
∗
→ X
∗
− e
−1
0
∧ (tX
∗
) (11.14)
APPENDIX D COMMON EQUATIONS 607
Table D.8: Convenient equations from the conformal model.
Requirement Formula Reference
Euclidean distance of normalized points p, q d
E
(p, q ) =
|−2p · q| from (13.2)
normalized point from Euclidean vector p p = o + p +
1
2
p
2
∞ (13.3)
normalized point from normalized sphere Σ p =Σ
∗
−
1
2
Σ
2
∞
normalized point from normalized flat point Φ p = oΦ+
1
2
(oΦ)
2
∞
two normalized points from point pair P p
±
= (P ∓
|P
2
|)
/
(−∞P) (14.13)
normalize point or dual sphere p p → p
/
(−∞ · p) Section 13.1.3
normalize flat F F →
F
/
((o ∧∞)F)
dual sphere of radius ρ at normalized point c Σ
∗
= c −
1
2
ρ
2
∞
Table 13.2
dual sphere through p,centerc Σ
∗
= p(c ∧∞) (14.4)
direct round at c,radiusρ Σ=(c +
1
2
ρ
2
∞) ∧
− c(
A
k
∞)
(14.9)
dual round at c,radiusρ Σ
∗
= (c −
1
2
ρ
2
∞) ∧
− c(
A
夹
k
∞)
(14.10)
dual plane normal vector n,supportδ Π
∗
= n − δn∞ Table 13.2
dual plane through p, normal n Π
∗
= p(n ∧∞)
dual midplane normalized points p, q Π
∗
= p − q (14.5)
direct flat at p, direction A
k
Π=p ∧ A
k
∧∞ (13.8)
dual flat at p, direction A
k
Π
∗
= −p(A
夹
k
∞) (13.9)
parameters of all blades flats, rounds, tangents, directions Table 14.1
translation of Euclidean element T
p
[A
k
] = −p(∞A
k
) = A
k
+ ∞(pA
k
) (13.7)
logarithm rigid body motion algorithm based on formula Figure 13.5
logarithm scaled rigid body motion algorithm based on formula Figure 16.5
conformal versors complete list Table 16.1
Bibliography
[1] Ablamowicz, R., P. Lounesto, and J. Parra (Eds). Clifford Algebras with Numeric
and Sy mbolic Computations. Boston: Birkh
¨
auser, 1996.
[2] Angl
`
es, P. Construction de revˆetements du group conform d’un espace vectorial
muni d’une “m
´
etrique” de type (p,q). Annales de l’Institut Henri Poincar
´
e, Section
A, Vol. XXXIII:33–51, 1980.
[3] Barnabei, M., A. Brini, and G. C. Rota. On the exterior calculus of invariant
theory. Journal of Algebra, 96:120–160, 1985.
[4] Bayro-Corrochano, E., and J. Lasenby. A unified language for computer vision
and robotics. In G. Sommer and J. J. Koenderink (Eds), Algebraic Frames for the
Perception-Action Cycle. Lecture Notes in Computer Science, 1315, Springer 1997.
[5] Beckmann, O., A. Houghton, P. Kelly, and M. Mellor. Run-time code generation
in C++ as a foundation for domain-specific optimisation. Lecture Notes in
Computer Scie nce, 3016:291–306, 2004.
[6] Bell, I., Ian Bell’s web pages on geometric algebra.
http://www.iancgbell.clara.net/maths/geoalg1.htm, 2005.
[7] Bouma, T. A., and G. Memowich. Invertible homogeneous versors are blades.
http://www.science.uva.nl/ga/publications, 2001.
[8] Bouma, T., L. Dorst, and H. G. J. Pijls. Geometric algebra for subspace operations.
Acta Mathemat icae Applicandae, pp. 73:285–300, 2002.
[9] Cox, H. On systems of circles and bicircular quartics. Quaterly Journal of Pure and
Applied Mathematics, 19:74–124, 1883.
[10] Cox, H. Application of Grassmann’s Ausdehnungslehre to properties of circles.
Quarterly Journal of Pure and Applied Mathematics, 25:1–71, 1891.
[11] Czarnecki, K., and U. W. Eisenecker. Generative Programming—Methods, Tools,
and Applications. Addison-Wesley, 2000.
[12] de Berg, M., M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational
Geometry: Algorithms and Applications, 2nd edition. Springer Verlag, 1998.
[13] DeRose, T. Coordinate-free geometric programming. Technical Report 89-09-16,
University of Washington, September 1989.
[14] Doran, C., and A. Lasenby. Physical applications of geometric algebra.
http://www.mrao.cam.ac.uk/
~
clifford/ptIIIcourse/, 1999.
609
610 BIBLIOGRAPHY
[15] Doran, C., and A. Lasenby. Geometric Algebra for Physicists. Cambridge University
Press, 2003.
[16] Dorst, L. Analyzing the behaviors of a car: a study in the abstraction of
goal-directed motion. IEEE Systems, Man and Cybernetics A, 28(6):811–822,
1998.
[17] Dorst, L. The inner products of geometric algebra. In J. Lasenby, L. Dorst, and
C. Doran (Eds), Applications of Geometric Algebra in Computer Science and
Engineering, pp. 35–46. Boston: Birkh
¨
auser, 2002.
[18] Dorst, L., and D. Fontijne. 3D Euclidean geometry through conformal geometric
algebra (a
GAViewer tutorial). http://www.science.uva.nl/ga/.
[19] Dorst, L., and S. Mann. Geometric algebra: a computation framework for
geometrical applications: part I. Computer Graphics and Applications,
22(3):24–31, May/June 2002.
[20] Eastwood, M. G., and P. W. Michor. Some remarks on the Pl
¨
ucker relations.
Rendiconti del Circolo Matematico di Palermo, II-63:85–88, 2000.
[21] Faugeras, O., and Q.-T. Luong. The Geometry of Multiple Views. MIT Press, 2001.
[22] Forder,H.G.The Calculus of Extension. Cambridge University Press, 1941.
[23] Goldman, R. Illicit expressions in vector algebra. ACM Transactions on Graphics,
4(3), July 1985.
[24] Goldman, R. On the algebraic and geometric foundations of computer graphics.
ACM Transactions on Graphics, 21(1), January 2002.
[25] Hartley, R. I., and A. Zisserman. Multiple View Geometry in Computer Vision, 2nd
edition. Cambridge University Press, 2003.
[26] Hestenes, D. Geometric calculus. A chapter of an unpublished book, available at
http://modelingnts.la.asu.edu/html/NFMP.html.
[27] Hestenes, D. The design of linear algebra and geometry. Acta Appl. Math,
23:65–93, 1991.
[28] Hestenes, D. Grassmann’s vision. In Hermann G
¨
unther Grassmann (1809–1877):
Visionary Mathematician, Scientist and Neohumanist Scholar, pp. 243–254.
Dordrecht: Kluwer Academic Publishers Group, 1996.
[29] Hestenes, D. New Foundations for Classical Mechanics, 2nd edition. Reidel, 2000.
[30] Hestenes, D. Point groups and space groups. In J. Lasenby, L. Dorst, and C. Doran
(Eds), Applications of Geometric Algebra in Computer Scie nce and Engineering,
pp. 3–34. Boston: Birkh
¨
auser, 2002.
[31] Hestenes, D., H. Li, and A. Rockwood. A unified algebraic framework for classical
geometry. In G. Sommer (Ed), Geometric Computing with Clifford Algebra.
Springer, 1999.
BIBLIOGRAPHY 611
[32] Hestenes, D., A. Rockwood, and H. Li. System for e ncoding and manipulating
models of objects, U.S. Patent 6,853,964, granted February 8, 2005.
[33] Hestenes, D., and G. Sobczyk. Clifford Algebra to Geometric Calculus. Reidel, 1984.
[34] Hildenbrand, D., D. Fontijne, W. Yusheng, M. Alexa, and L. Dorst. Competitive
run-time performance for inverse kinematics algorithms using conformal
geometric algebra. In Proceedings of Eurographics 2006, Vienna, 2006.
[35] Koenderink, J. J. A Generic Framework for Image Geometry. In J. Lasenby,
L. Dorst, and C. Doran, (Ed), Applications of Geometric Algebra in Computer
Science and Engineering, pp. 319–332. Boston: Birkh
¨
auser, 2002.
[36] Lasenby, J., W. J. Fitzgerald, C. J. L. Doran, and A. N. Lasenby. New geometric
methods for computer vision. International Journal of Computer Vision,
36(3):191–213, 1998.
[37] Lasenby, J., R. Lasenby, A. Lasenby, and R. Wareham. Higher dimensional fractals
in geometric algebra. Technical Report CUED/F-INFENG/TR.556, Cambridge
University Engineering Department, 2006.
[38] Lasenby, J., and A. Stevenson. Using geometric algebra for optical motion
capture. In E. Bayro-Corrochano and G. Sobcyzk, (Eds), Applied Clifford Algebras
in Computer Science and Engineering. Birkh
¨
auser, 2000.
[39] Lounesto, P. Marcel Riesz’s work on Clifford algebras. In E. F. Bolinder and
P. Lounesto (Eds), Clifford Numbers and Spinors, pp. 119–241. Kluwer Academic,
1993.
[40] Lounesto, P., R. Mikkola, and V. Vierros. CLICAL user manual: complex number,
vector space, and Clifford algebra calculator for MS-DOS personal computers.
Technical Report Institute of Mathematics Research Reports A248, Helsinki
University of Technology, 1987.
[41] Lounesto, P. Clifford algebras and spinors. London Mathematical Society Lecture
Note Series 239, 1997.
[42] Mann, S., and L. Dorst. Geometric algebra: a computation framework for
geometrical applications: part II. Computer Graphics and Applications,
22(4):58–67, July/August 2002.
[43] Mann, S., L. Dorst, and T. Bouma. The making of GABLE, a geometric algebra
learning environment in Matlab. In E. Bayro-Corrochano and G. Sobczyk (Eds),
Geometric Algebra with Applications in Science and Engineering, chapter 24, pp.
491–511. Birkh
¨
auser, 2001.
[44] Mann, S., N. Litke, and T. DeRose. A coordinate free geometry ADT. Technical
Report CS-97-15, University of Waterloo, 1997.
[45] Mann, S., and A. Rockwood. Using geometric algebra to compute singularities in
3D vector fields. In IEEE Visualization 2002 Proceedings, pp. 283–289, 2002.
[46] Needham, T. Visual Complex Analysis. Oxford: Clarendon Press, 1997.
612 BIBLIOGRAPHY
[47] Perwass, C., CLU: Clifford algebra library and utilities.
http://www.perwass.de/cbup/clu.html.
[48] Perwass, C., and W. F
¨
orstner. Uncertain Geometry with Circles, Spheres and
Conics, volume 31 of Computational Imaging and Vision, pp. 23–41.
Springer-Verlag, 2006.
[49] Porteous, I. R. Topological Geometry. Cambridge: Cambridge University Press,
1981.
[50] Porteous, I. R. Geometric Differentiation for the Intelligence of Curves and Surfaces.
Cambridge: Cambridge University Press, 1994.
[51] Porteous, I. R. Clifford Algebras and the Classical Groups. Cambridge: Cambridge
University Press, 1995.
[52] Riesz, M. Clifford Numbers and Spinors. Kluwer Academic, 1993.
[53] Rosenhahn, B., and G. Sommer. Pose estimation in conformal geometric algebra,
part I: the stratification of mathematical spaces. Journal of Mathemat ical Imaging
and Vision (JMIV), 22:27–48, 2005.
[54] Rosenhahn, B., and G. Sommer. Pose estimation in conformal geometric algebra,
part II: real-time pose estimation using extended feature concepts. Journal of
Mathematical Imaging and Vision (JMIV), 22:49–70, 2005.
[55] Semple, J. G., and G. T. Kneebone. Algebraic Projective Geometry. Oxford
University Press, 1952.
[56] Shoemake, K. Animating rotation with quaternion curves. In Proceedings of ACM
SIGGRAPH, volume 19:3, pp. 245–254, 1985.
[57] Shoemake, K. Pl
¨
ucker coordinate tutorial. Ray Tracing News, 11(1), July 1998.
[58] Shreiner, D., M. Woo, J. Neider, T. Davis, and the OpenGL Architecture Review
Board. OpenGL Programming Guide: The Official Guide to Learning OpenGL,
Version 1.4, Fourth Edition. Addison-Wesley, 2003.
[59] Stewart. I. Hermann Grassmann was right. Nature, 321:6065, 1986.
[60] Stolfi, J. Oriented Projective Geometr y. Academic Press, 1991.
[61] Suter, J. Clifford (software), 2003. Used to be available at
http://www.jaapsuter.com.
[62] Valkenburg, R. J., and N. S. Alwesh, Calibration of Target Positions using the
Conformal Model and Geometric Algebra, Image and Vision Computing, New
Zealand, 241–246, 2005.
[63] Zhang, Z. A flexible new technique for camera calibration. IEEE Transactions on
Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000.
[64] Zhao, Y., R. J. Valkenburg, R. Klette, and B. Rosenhahn, Target Calibration and
Tracking using Conformal Geometric Algebra, IEEE Pacific-Rim Symposium on
Image and Video Technology, 2006.
Index
∧,50
∗, 67, 154
, 73, 154
, 77,154
•, 589
•
H
, 590
×,86
x, 215
M, 538
∗
,80
−∗
,81
夹
, 289, 374
,47
,47
grade(),47
k
,47
⊆ ,43
∩, ∪, 127
f
∗
, 111
f, 109
det(f), 106
∂, 221–36
e
0
, 272
e,
e , 361
o, ∞, 359
R
n
, 586
R
n,0
, 586
R
n+1,1
, 586
R
p,q
, 586
E
n
, 356
k
R
n
,47
R
n
,47
0-blades, 37, 38, 44
2-blades, 29
angle between subspaces, 69
lines as, 278–83
pure rotors as exponentials, 183–84
reshapability, 31–32, 279
squared norm of subspace, 68
squaring to scalars, 184
See also Blades
3-blades, planes as, 283–85
3-D cross product, 86–89, 94–95
3-D rotations
multiple, 179
visualizing, 179–80
see also Quaternions
3-D rotors
computing with, 256–60
determining, 256–58
logarithm of, 258–59
rotation interpolation, 259–60
3-D space
angular relationships in, 251–54
defined, 8
Grassmann algebra for, 48
quadvectors in, 37
rotors in, 181
translations in, 219
A
Addition
bivector, 32–33, 199
in linear operations implementation,
522
of lines, 282–83, 324–25
of points, 276–78
products distributive over, 197
Adjoint
as derivative, 233–34
of linear transformation, 108–9
Affine combinations
in conformal model, 447–49
of circles, 448–49
defined, 277
example, 463
of flats, 448
of points, 307–8, 463–64
Affine distance ratio, 299
Affine transformations, 299, 306–8, 334
defined, 306
Euclidean motions as, 307
See also Transformations
Algebra of directions, 247–70, 498
Algebra specification, 546–47
constants, 547
dimension, 546
metric, 546–47
specialized types, 547
Angles
cosine law of, 254
rotor determination from, 256–57
between subspaces, 68–71
Angular relationships, 248–56
in 3-D, 251–54
geometry of planar triangles, 249–51
rotation groups and, 254–56
between three directions, 252–53
two bivectors, 251–52
two vectors, 251
vector and bivector, 252
Antisymmetry, 29, 51, 143
Applications
all kinds of vectors, 451–55
fitting a sphere to points, 417–20
homogeneous model, 327–54
homogeneous Pl
¨
ucker coordinates,
328–36
imaging by multiple cameras, 336–46
intersecting planar lines, 41
inversion, 467, 468–69
kinematics, 420–26
radius of curvature of planar curve,
223–24
ray-tracing, 557–81
reciprocal frames, 89–90
613
614 INDEX
solving linear equations, 39–40
tilting a mirror, 228–30
Voronoi cell analysis, 455–60
Voronoi diagrams, 415–17
Area elements, 28, 30
in 3-D, 30
oriented, 251
standard, 30
Associativity
geometric product of multivectors,
148
geometric product of vectors, 144
of outer product, 35–36
Attitudes
defined, 43
directions, 248
homogeneous planes, 27–28
homogeneous spaces, 133
lines, 26
points, 38, 275–76
sine measure, 133
volumes, 33
Automatic code generator, 543
B
Backface culling, 57–60, 430–31
Barycentric coordinates, 316
Base space
defined, 272
elements, 274
projective transformations, 299
pseudoscalar, 289
Basis blades, 511–19
bitmap representation, 512
Clifford conjugate, 519
commutator product of, 518
defined, 512
geometric product, 516–17
grade-dependent signs on, 518–19
grade involution, 519
list of, 526–27
metric products, 518
outer product, 513–15
product evaluations, 526
reversion, 519
unit, representing with bitmaps,
512–13
See also Blades
Basis vectors, geomet ric product,
144–45
Benchmarks, 554–56
Bilinear form, 66, 148, 585
Bitwise boolean operators, 513
Bivector addition, 199
in 2-D space, 32–33
in 3-D space, 34
geometrical representation, 32
visualizing, 32–33
Bivector angle, 174
Bivectors
in angular relationships, 251–52
defined, 29
drawing, 57
drawing code, 58
drawing screenshot, 59
duality of, 82
geometric reshapability, 31–32,
39–40, 41
representations, 32
rotors as exponentials, 185–87
visualizing, 31–32
Blades
basis, 511–19
defined, 44, 195
delta product, 537, 538
direction, 408–9
even unit, 195
factorization, 533–35, 554
grade-reducing product between, 72
intersection, 127
inverses, 79–80, 155, 529–30, 554
invertible, 195
join of, 536–40, 554
ladder of, 51
linear transformations, 101–7, 302
location, 409
meet of, 536–40, 554
notational convention, 47
null, 80, 587
orientation, 408–9
rounds as, 410–15
squared size, 408
union of, 127
in versor product, 195
test, 532–33,
vs multivectors, 46, 198
weight, 43, 408–9
See also k-blades
Boolean operators, 511, 513
Bounding sphere
computing, 563–64
test, 577
BSP t ree
construction, 564–65
traversing, 577–79
C
Cameras
calibration, 260–63, 268–70, 351–54
multiple, imaging by, 336–46
orbiting, 572
pinhole, 337–39, 347
rays, 575–76
rotating, 571–72
stereo vision and, 340–42
translating, 570–71
Carriers, 445
Chain rule, 234
Chasles’ theorem, 381, 383
Circles
circum-, 457–59
direct, 402
affine combination of, 448–49
creating, 432
dual, 399
Euclidean, intersection as planes,
414–15
intersection, 414
oriented, 402
through points, 2, 404
reflection, 468–69
representation, 412–13
rotation, 479
See also Rounds
Clifford, W.K., 6, 143
Classification
conformal blades, 407
homogeneous blades, 291
multivector, 532–33, 554
Clifford algebra, 141
4-D representation space, 336
defined, 198
dormant results, 199
geometric algebra versus, 198–99
matrix representation, 506–7
of spaces, 507
Clifford conjugation, 53, 519, 604
Clifford group, 196
Coincident lines, 294
Color space conversion, 95–98
code, 97