Applications of geometric algebra in computer science and engineering:
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Boston ; Basel ; Berlin
Birkhäuser
2002
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIV, 478 S. Ill., graph. Darst. : 24 cm |
| ISBN: | 3764342676 0817642676 |
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| 245 | 1 | 0 | |a Applications of geometric algebra in computer science and engineering |c Leo Dorst ... (ed.) |
| 264 | 1 | |a Boston ; Basel ; Berlin |b Birkhäuser |c 2002 | |
| 300 | |a XXIV, 478 S. |b Ill., graph. Darst. : 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Informatik | |
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Computer science |x Mathematics | |
| 650 | 4 | |a Engineering mathematics | |
| 650 | 4 | |a Geometry, Algebraic | |
| 650 | 0 | 7 | |a Geometrische Algebra |0 (DE-588)4156707-9 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)1071861417 |a Konferenzschrift |y 2001 |z Cambridge |2 gnd-content | |
| 689 | 0 | 0 | |a Geometrische Algebra |0 (DE-588)4156707-9 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Dorst, Leo |d 1958- |e Sonstige |0 (DE-588)114432058 |4 oth | |
| 856 | 4 | 2 | |m DNB Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009843855&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-009843855 | |
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CONTENTS
PREFACE
V
CONTRIBUTORS
VII
I
ALGEBRA
AND
GEOMETRY
1
1
POINT
GROUPS
AND
SPACE
GROUPS
IN
GEOMETRIC
ALGEBRA
DAVID
HESTENES
3
1.1
INTRODUCTION
.
3
1.2
POINT
GROUPS
IN
TWO
DIMENSIONS
.
5
1.3
POINT
GROUPS
IN
THREE
DIMENSIONS
.
10
1.4
THE
32
CRYSTAL
CLASSES
AND
7
CRYSTAL
SYSTEMS
.
17
1.5
HOMOGENEOUS
EUCLIDEAN
GEOMETRY
.
22
1.6
SYMMETRIES
FROM
REFLECTIONS
.
25
1.7
THE
SPACE
GROUPS
.
27
1.7.1
PLANAR
SPACE
GROUPS
.
29
1.7.2
3D
SPACE
GROUPS
.
32
2
THE
INNER
PRODUCTS
OF
GEOMETRIC
ALGEBRA
LEO
DORST
35
2.1
THE
PRODUCT
STRUCTURE
OF
GEOMETRIC
ALGEBRA
.
35
2.2
THE
BASIC
PRODUCTS
OF
GEOMETRIC
ALGEBRA
.
36
2.2.1
GEOMETRIC
PRODUCT
.
36
2.2.2
OUTER
PRODUCT;
BLADES
.
37
2.2.3
SCALAR
PRODUCT:
METRIC
PROPERTIES
.
37
2.2.4
CONTRACTIONS
.
38
2.2.5
RELATIONSHIP
OF
THE
CONTRACTION
TO
THE
INNER
PRODUCT
.
39
2.3
UNDERSTANDING
THE
CONTRACTION
.
40
2.3.1
DEFINING
AXIOMS
.
40
2.3.2
FAMOUS
FORMULAS
.
40
2.3.3
GEOMETRIC
INTERPRETATION
.
41
2.4
PROJECTION
.
42
2.5
MEET
AND
JOIN
.
43
2.6
LINEAR
TRANSFORMATIONS
AS
'
INNERMORPHISMS
'
.
44
2.6.1
OUTERMORPHISMS
AND
ADJOINT
TRANSFORMATIONS
.
44
2.6.2
COVARIANCE
OF
INNER
PRODUCT
FORMULAS
.
44
2.7
REPLACING
THE
INNER
PRODUCT
.
45
XIV
CONTENTS
3
UNIFICATION
OF
GRASSMANN
'
S
PROGRESSIVE
AND
REGRESSIVE
PRODUCTS
USING
THE
PRINCIPLE
OF
DUALITY
STEPHEN
BLAKE
47
3.1
INTRODUCTION
.
47
3.2
POINTS
.
48
3.3
GRASSMANN
'
S
ALGEBRA
OF
POINTS
.
49
3.4
HYPERPLANES
AND
COORDINATES
FOR
POINTS
.
51
3.5
POINTS
AND
COORDINATES
FOR
HYPERPLANES
.
51
3.6
REPRESENTING
REGIONS
BY
POINTS
OR
HYPERPLANES
.
52
3.7
RULE
OF
THE
MIDDLE
FACTOR
.
53
3.8
FUNDAMENTAL
FORMULAE
OF
WHITEHEAD
'
S
ALGEBRA
.
54
3.9
EXAMPLE:
THE
HARMONIC
SECTION
THEOREM
.
55
4
FROM
UNORIENTED
SUBSPACES
TO
BLADE
OPERATORS
TIMAEUS
A.
BOUMA
59
4.1
INTRODUCTION
.
59
4.1.1
A
REVIEW
OF
GEOMETRIC
ALGEBRA
.
59
4.2
DEFINITIONS
.
61
4.2.1
PROJECTION
OPERATORS
.
61
4.2.2
DELTA
PRODUCT
.
61
4.2.3
LIFT
.
62
4.2.4
MEET
AND
JOIN
.
62
4.3
PROJECTION
OPERATORS
.
63
4.3.1
PROJECTION
AND
THE
INNER
PRODUCT
.
63
4.3.2
FUNDAMENTAL
THEOREM
OF
PROJECTION
OPERATORS
.
63
4.3.3
SPECIAL
CASES
.
63
4.4
THE
MEET
AND
JOIN
FOR
PROJECTION
OPERATORS
.
64
4.4.1
THE
MEET
AND
JOIN
WHEN
AB
IS
A
BLADE
.
64
4.4.2
THE
MEET
AND
JOIN
WHEN
AB
IS
NOT
A
BLADE
.
64
4.5
FROM
UNORIENTED
SUBSPACES
TO
BLADE
OPERATORS
.
65
4.5.1
THE
BLADE
CORRESPONDENCE
.
65
4.5.2
LIFT
TO
A
EUCLIDEAN
METRIC
.
66
4.6
CONCLUSION
.
66
5
AUTOMATED
THEOREM
PROVING
IN
THE
HOMOGENEOUS
MODEL
WITH
CLIFFORD
BRACKET
ALGEBRA
HONGBO
LI
69
5.1
CLIFFORD
BRACKET
ALGEBRA
.
71
5.2
SOME
BRACKET
EXPRESSIONS
.
72
5.3
THEOREM
PROVING
WITH
CLIFFORD
BRACKET
ALGEBRA
.
74
CONTENTS
XV
6
ROTATIONS
IN
N
DIMENSIONS
AS
SPHERICAL
VECTORS
W.
E.
BAYLIS
AND
S.
HADI
79
6.1
INTRODUCTION
.
79
6.2
S
'
1
"
1
MODEL
.
81
6.2.1
SPINOR
ROTATIONS
AND
SPHERICAL
VECTORS
.
81
6.2.2
COMPOSING
ROTATIONS
BY
ADDING
SPHERICAL
VECTORS
.
84
6.3
EXAMPLES
.
87
6.3.1
ROTATIONS
IN
E
3
.
87
6.3.2
ROTATIONS
IN
E
4
.
88
6.4
CONCLUSIONS
.
89
7
GEOMETRIC
AND
ALGEBRAIC
CANONICAL
FORMS
NEIL
GORDON
91
7.1
INTRODUCTION
.
91
7.2
CLIFFORD
ALGEBRAS
AND
FINITE
GEOMETRY
.
91
7.3
GEOMETRIC
STRUCTURES
AND
ALGEBRAIC
FORMS
.
93
7.4
GEOMETRIC
CANONICAL
FORMS,
ALGEBRAIC
CANONICAL
FORMS
AND
COMPUTER
ALGEBRA
.
95
7.5
CONCLUSION
.
96
8
FUNCTIONS
OF
CLIFFORD
NUMBERS
OR
SQUARE
MATRICES
JOHN
SNYGG
99
8.1
INTRODUCTION
.
99
8.2
USING
THE
MINIMAL
POLYNOMIAL
.
99
8.3
USING
THE
CHARACTERISTIC
POLYNOMIAL
.
104
8.4
CONCLUDING
REMARKS
.
106
9
COMPOUND
MATRICES
AND
PFAFFIANS:
A
REPRESENTATION
OF
GEOMETRIC
ALGEBRA
UWE
PRELLS,
MICHAEL.
I.
FRISWELL,
AND
SEAMUS
D.
GARVEY
109
9.1
GRASSMANN
ALGEBRA
AND
COMPOUND
MATRICES
.
109
9.2
PFAFFIANS
AND
THEIR
GENERALISATION
.
112
9.3
REPRESENTATION
OF
THE
CLIFFORD
ALGEBRA
C^
N
(IF)
.
114
9.4
CLIFFORD
POWERS
.
116
10
ANALYSIS
USING
ABSTRACT
VECTOR
VARIABLES
FRANK
SOMMEN
119
10.1
INTRODUCTION
.
119
10.2
THE
ALGEBRA
OF
ABSTRACT
VECTOR
VARIABLES
.
119
10.3
ANALYSIS
IN
ABSTRACT
VECTOR
VARIABLES
.
123
10.4
CONCLUSION
.
127
XVI
CONTENTS
11
A
MULTIVECTOR
DATA
STRUCTURE
FOR
DIFFERENTIAL
FORMS
AND
EQUATIONS
JEFFREY
A.
CHARD
AND
VADIM
SHAPIRO
129
12
JET
BUNDLES
AND
THE
FORMAL
THEORY
OF
PARTIAL
DIFFERENTIAL
EQUATIONS
RICHARD
BAKER
AND
CHRIS
DORAN
133
12.1
FIBRE
BUNDLES
AND
SECTIONS
.
133
12.2
JET
BUNDLES
.
134
12.3
DIFFERENTIAL
FUNCTIONS
AND
FORMAL
DERIVATIVES
.
135
12.4
PROLONGATION
OF
SECTIONS
.
135
12.5
DIFFERENTIAL
EQUATIONS
AND
SOLUTIONS
.
136
12.6
PROLONGATION
AND
PROJECTION
OF
DIFFERENTIAL
EQUATIONS
.
137
12.7
POWER
SERIES
SOLUTIONS
.
137
12.8
INTEGRABILITY
CONDITIONS
.
138
12.9
INVOLUTIVE
DIFFERENTIAL
EQUATIONS
.
139
12.10
CARTAN-KURANISHI
COMPLETION
.
141
12.11
CONCLUSION
.
142
13
IMAGINARY
EIGENVALUES
AND
COMPLEX
EIGENVECTORS
EXPLAINED
BY
REAL
GEOMETRY
ECKHARD
M.S.
HITZER
145
13.1
INTRODUCTION
.
145
13.2
TWO
REAL
DIMENSIONS
.
146
13.2.1
COMPLEX
TREATMENT
.
146
13.2.2
REAL
EXPLANATION
.
147
13.3
THREE
REAL
DIMENSIONS
.
149
13.3.1
COMPLEX
TREATMENT
OF
THREE
DIMENSIONS
.
149
13.3.2
REAL
EXPLANATION
FOR
THREE
DIMENSIONS
.
150
13.4
FOUR
EUCLIDEAN
DIMENSIONS
-
COMPLEX
TREATMENT
.
153
14
SYMBOLIC
PROCESSING
OF
CLIFFORD
NUMBERS
IN
C++
JOHN
P.
FLETCHER
157
14.1
INTRODUCTION
.
157
14.2
SYMBOLICC++
.
158
14.2.1
OPERATOR
AND
TYPE
DEFINITIONS
.
158
14.2.2
TYPE
CONVERSION
.
159
14.2.3
OUTPUT
.
160
14.2.4
NONCOMMUTATIVE
ARITHMETIC
.
160
14.2.5
EXTENDING
SYMBOLICC++
.
160
14.3
CLIFFORD
ALGEBRA
EXAMPLES
.
161
14.3.1
QUATERNION
STARTING
POINT
.
161
14.3.2
CLIFFORD
(2)
.
161
14.3.3
CLIFFORD
(3)
.
163
CONTENTS
XVII
14.3.4
CLIFFORD
(2,2)
.
163
14.4
INTERACTIVE
INTERFACE
TO
TEL
USING
SWIG
.
164
14.5
STORING
ALGEBRA
USING
XML
AND
E4GRAPH
.
165
14.6
CONCLUSIONS
.
166
15
CLIFFORD
NUMBERS
AND
THEIR
INVERSES
CALCULATED
USING
THE
MATRIX
REPRESENTATION
JOHN
P.
FLETCHER
169
15.1
INTRODUCTION
.
169
15.2
CLIFFORD
BASIS
MATRIX
THEORY
.
170
15.3
CALCULATION
OF
THE
INVERSE
OF
A
CLIFFORD
NUMBER
.
172
15.3.1
EXAMPLE
1:
CLIFFORD
(2)
.
173
15.3.2
EXAMPLE
2:
CLIFFORD
(3)
.
173
15.3.3
EXAMPLE
3:
CLIFFORD
(2,2)
.
175
15.4
CONCLUSION
.
178
16
A
TOY
VECTOR
FIELD
BASED
ON
GEOMETRIC
ALGEBRA
ALYN
ROCKWOOD
AND
SHOEB
BINDERWALA
179
16.1
INTRODUCTION
.
179
16.2
THE
CONJUGATE
FIELD
METHOD
.
181
16.3
COMPARISON
.
183
16.4
IMPLEMENTATION
.
184
17
QUADRATIC
TRANSFORMATIONS
IN
THE
PROJECTIVE
PLANE
GEORGI
GEORGIEV
187
17.1
INTRODUCTION
.
187
17.2
QUADRATIC
INVOLUTION
IN
THE
PROJECTIVE
PLANE
.
187
17.3
QUADRATIC
INVOLUTION
WHICH
LEAVES
AN
INVARIANT
ELLIPSE
.
.
.
189
18
ANNIHILATORS
OF
PRINCIPAL
IDEALS
IN
THE
GRASSMANN
ALGEBRA
CEMAL
KOC
AND
SONGUL
ESIN
193
II
APPLICATIONS
TO
PHYSICS
195
19
HOMOGENEOUS
RIGID
BODY
MECHANICS
WITH
ELASTIC
COUPLING
DAVID
HESTENES
AND
ERNEST
D.
FASSE
197
19.1
INTRODUCTION
.
197
19.2
HOMOGENEOUS
EUCLIDEAN
GEOMETRY
.
199
19.3
RIGID
DISPLACEMENTS
.
202
19.4
KINEMATICS
.
206
19.5
DYNAMICS
.
208
19.6
ELASTIC
COUPLING
.
209
19.7
CONCLUSIONS
.
210
XVIII
CONTENTS
20
ANALYSIS
OF
ONE
AND
TWO
PARTICLE
QUANTUM
SYSTEMS
USING
GEOMETRIC
ALGEBRA
RACHEL
PARKER
AND
CHRIS
DORAN
213
20.1
INTRODUCTION
.
213
20.2
SINGLE-PARTICLE
PURE
STATES
.
214
20.3
2-PARTICLE
SYSTEMS
.
214
20.3.1
SCHMIDT
DECOMPOSITION
.
215
20.3.2
THE
DENSITY
MATRIX
.
216
20.4
GEOMETRIC
ALGEBRA
.
217
20.4.1
SINGLE-PARTICLE
SYSTEMS
.
217
20.4.2
2-PARTICLE
SYSTEMS
.
219
20.4.3
2-PARTICLE
OBSERVABLES
.
222
20.4.4
THE
DENSITY
MATRIX
.
223
20.4.5
EXAMPLE
-
THE
SINGLET
STATE
.
224
21
INTERACTION
AND
ENTANGLEMENT
IN
THE
MULTIPARTICLE
SPACETIME
ALGEBRA
TIMOTHY
F.
HAVEL
AND
CHRIS
J.
L.
DORAN
227
21.1
THE
PHYSICS
OF
QUANTUM
INFORMATION
.
227
21.2
THE
MULTIPARTICLE
SPACETIME
ALGEBRA
.
229
21.3
TWO
INTERACTING
QUBITS
.
230
21.3.1
THE
PROPAGATOR
.
233
21.3.2
OBSERVABLES
.
234
21.4
LAGRANGIAN
ANALYSIS
.
237
21.4.1
SINGLE-PARTICLE
SYSTEMS
.
237
21.4.2
TWO-PARTICLE
INTERACTIONS
.
238
21.4.3
SYMMETRIES
AND
NOETHER
'
S
THEOREM
.
239
21.5
THE
DENSITY
OPERATOR
.
240
21.5.1
AN
EXAMPLE
OF
INFORMATION
DYNAMICS
.
242
21.5.2
TOWARDS
QUANTUM
COMPLEXITY
AND
DECOHERENCE
.
.
244
22
LAWS
OF
REFLECTION
FROM
TWO
OR
MORE
PLANE
MIRRORS
IN
SUCCESSION
MIKE
DERDME
249
22.1
INTRODUCTION
.
249
22.2
SINGLE
REFLECTION
IN
3D
EUCLIDEAN
SPACE
.
250
22.3
TWO
SUCCESSIVE
REFLECTIONS
.
251
22.3.1
DEFINITIONS
.
252
22.3.2
LAWS
OF
DOUBLE
REFLECTION
.
252
22.3.3
THE
SPREAD
ANGLE
FOR
TWO
REFLECTIONS
.
253
22.3.4
PERSISTENCE
FOR
TWO
REFLECTIONS
.
253
22.3.5
REFLECTION
IN
THE
PLANE
OF
MIRROR
NORMALS
-
2D
REFLECTION
.
253
22.3.6
RETRO-REFLECTION
SOLUTIONS
FOR
TWO
REFLECTIONS
.
253
22.4
THREE
SUCCESSIVE
REFLECTIONS
.
253
CONTENTS
XIX
22.4.1
TWO
IMPORTANT
SPECIAL
CONFIGURATIONS
EXIST:
.
254
22.4.2
LAWS
OF
TRIPLE
REFLECTION
.
255
22.4.3
THE
SPREAD
ANGLE
FOR
THREE
REFLECTIONS
.
255
22.4.4
PERSISTENCE
FOR
THREE
REFLECTIONS
.
256
22.4.5
REFLECTION
IN
THE
PLANE
OF
ALL
MIRROR
NORMALS
-
2D
REFLECTION
.
256
22.4.6
RETRO-REFLECTIVE
SOLUTIONS
FOR
THREE
REFLECTIONS
.
.
.
256
22.4.7
CONSTRAINTS
ON
RETRO-REFLECTION
FROM
THREE
ONE-SIDED
MIRRORS
.
256
22.5
AN
ARBITRARY
NUMBER
OF
SUCCESSIVE
REFLECTIONS
.
257
22.6
CONCLUSIONS
.
258
23
EXACT
KINETIC
ENERGY
OPERATORS
FOR
POLYATOMIC
MOLECULES
JANNE
PESONEN
261
23.1
INTRODUCTION
.
261
23.2
KINETIC
ENERGY
OPERATOR
.
262
23.2.1
VIBRATIONAL
DEGREES
OF
FREEDOM
.
264
23.2.2
ROTATIONAL
DEGREES
OF
FREEDOM
.
265
23.3
COMPARISON
TO
OTHER
APPROACHES
.
268
24
GEOMETRY
OF
QUANTUM
COMPUTING
BY
HAMILTONIAN
DYNAMICS
OF
SPIN
ENSEMBLES
T.
SCHULTE-HERBRUGGEN,
K.
HUPER,
U.
HELMKE,
AND
S.J.
GLASER
271
24.1
FROM
HAMILTONIAN
QUANTUM
DYNAMICS
TO
C-NUMERICAL
RANGES
.
271
24.1.1
QUANTUM
MECHANICS
OF
ELEMENTARY
AND
JOINT
SYSTEMS
.
271
24.1.2
REDUCED
DESCRIPTIONS
OF
ENSEMBLES
.
272
24.1.3
RELATION
TO
THE
C-NUMERICAL
RANGE
.
273
24.2
UNITARY
CONTROLLABILITY
OF
SPIN
ENSEMBLES
.
274
24.3
FROM
GEOMETRY
TO
ENTROPY
AND
VICE
VERSA
.
276
24.3.1
GENERALISED
ANGLES
BETWEEN
STATES
OR
MATRICES
.
.
.
276
24.3.2
EUCLIDEAN
DISTANCES
BETWEEN
STATES
OR
MATRICES
.
.
.
277
24.3.3
ENTROPIES
VS
NORMS
AND
EUCLIDEAN
DISTANCES
.
277
24.4
GRADIENT
FLOW
MINIMISING
EUCLIDEAN
DISTANCE
.
279
24.5
A
CAVEAT
ON
ENTROPY
MEASURES
IN
NMR
.
280
25
IS
THE
BRAIN
A
'
CLIFFORD
ALGEBRA
QUANTUM
COMPUTER
'
?
V.
LABUNETS,
E.
RUNDBLAD,
AND
J.
ASTOLA
285
25.1
INTRODUCTION
.
285
25.2
CLIFFORD
ALGEBRA
AS
A
MODEL
OF
GEOMETRICAL
SPACE
.
287
25.3
CLIFFORD
ALGEBRA
AS
A
MODEL
OF
PERCEPTUAL
SPACE
.
288
25.4
HYPERCOMPLEX-VALUED
INVARIANTS
.
291
25.5
FAST
CALCULATION
ALGORITHMS
.
292
XX
CONTENTS
26
A
HESTENES
SPACETIME
ALGEBRA
APPROACH
TO
LIGHT
POLARIZATION
QURINO
M.
SUGON
AND
DANIEL
MCNAMARA
297
26.1
INTRODUCTION
.
297
26.2
ELECTROMAGNETIC
WAVE
.
298
26.3
THE
ENERGY-MOMENTUM
CLIFFOR
.
300
26.4
THE
POINCARE
SPHERE,
STOKES
PARAMETERS
AND
COHERENCY
MATRICES
.
301
26.5
POLARIZATION
STATES
.
303
26.6
SUMMARY
.
305
27
QUATERNIONS,
CLIFFORD
ALGEBRA
AND
SYMMETRY
GROUPS
PATRICK
R.
GIRARD
307
27.1
INTRODUCTION
.
307
27.2
CLIFFORD
ALGEBRAS
.
307
27.2.1
DEFINITIONS
AND
THEOREM
.
307
27.2.2
CLIFFORD
ALGEBRA
(C
4
OVER
R)
.
308
27.3
THE
LORENTZ
GROUP
.
309
27.3.1
PSEUDO-EUCLIDEAN
SPACE
.
309
27.3.2
RIEMANNIAN
SPACE
.
310
27.4
THE
CONFORMAL
GROUP
.
311
27.5
DIRAC
ALGEBRA
(C4
OVER
C)
.
312
27.5.1
DIRAC
'
S
EQUATION
.
312
27.5.2
UNITARY
AND
SYMPLECTIC
GROUPS
.
312
27.6
CONCLUSION
.
313
III
COMPUTER
VISION
AND
ROBOTICS
317
28
A
GENERIC
FRAMEWORK
FOR
IMAGE
GEOMETRY
JAN
J.
KOENDERINK
319
28.1
INTRODUCTION
.
319
28.2
THE
NATURAL
INTENSITY
SCALE
.
320
28.3
THE
SIMILARITIES
OF
IMAGE
SPACE
.
321
28.4
GEOMETRIES
OF
THE
PICTURE\AND
NORMAL
PLANES
COMPARED
.
.
322
28.4.1
THE
DIFFERENTIAL
GEOMETRY
OF
THE
NORMAL
PLANES
.
.
324
28.5
GEOMETRY
OF
IMAGE
SPACE
.
324
28.6
DIFFERENTIAL
GEOMETRY
OF
IMAGES
.
325
28.6.1
CURVES
.
326
28.6.2
SURFACES
.
327
28.7
RELATION
WITH
SCALE
SPACE
STRUCTURE
.
329
28.8
CONCLUSIONS
.
329
CONTENTS
XXI
29
COLOR
EDGE
DETECTION
USING
ROTORS
EDUARDO
BAYRO-CORROCHANO
AND
SANDINO
FLORES
333
29.1
INTRODUCTION
.
333
29.1.1
ROTORS
.
334
29.2
ROTOR
EDGE
DETECTOR
.
335
29.3
MODIFIED
ROTOR
EDGE
DETECTOR
.
336
29.4
CONCLUSION
.
339
30
NUMERICAL
EVALUATION
OF
VERSORS
WITH
CLIFFORD
ALGEBRA
CHRISTIAN
B.
U.
PERWASS
AND
GERALD
SOMMER
341
30.1
INTRODUCTION
.
341
30.2
THEORY
.
342
30.3
IMPLEMENTATION
.
346
30.4
EXPERIMENTS
.
347
30.5
CONCLUSION
.
349
31
THE
ROLE
OF
CLIFFORD
ALGEBRA
IN
STRUCTURE-PRESERVING
TRANSFORMATIONS
FOR
SECOND-ORDER
SYSTEMS
SEAMUS
D.
GARVEY,
MICHAEL
I.
FRISWELL,
AND
UWE
PRELLS
351
31.1
THE
CHARACTERISTIC
BEHAVIOUR
OF
SECOND-ORDER
SYSTEMS
.
.
351
31.2
INITIAL
INDICATIONS
OF
A
ROLE
FOR
CL?
.
352
31.3
STRUCTURE
PRESERVING
TRANSFORMATIONS
FOR
SECOND-ORDER
SYSTEMS
.
356
32
APPLICATIONS
OF
ALGEBRA
OF
INCIDENCE
IN
VISUALLY
GUIDED
ROBOTICS
EDUARDO
BAYRO-CORROCHANO,
PERTTI
LOUNESTO,
AND
LEO
REYES
LOZANO
361
32.1
ALGEBRA
OF
INCIDENCE
.
361
32.1.1
INCIDENCE
RELATIONS
IN
THE
AFFINE
N-PLANE
.
362
32.1.2
INCIDENCE
RELATIONS
IN
THE
AFFINE
3-PLANE
.
364
32.1.3
GEOMETRIC
CONSTRAINTS
AS
INDICATORS
.
365
32.2
RIGID
MOTION
IN
THE
AFFINE
PLANE
.
366
32.3
APPLICATION
TO
ROBOTICS
.
367
32.3.1
INVERSE
KINEMATIC
COMPUTING
.
367
32.3.2
ROBOT
MANIPULATION
GUIDANCE
.
370
32.3.3
CHECKING
FOR
A
CRITICAL
CONFIGURATION
.
371
33
MONOCULAR
POSE
ESTIMATION
OF
KINEMATIC
CHAINS
BODO
ROSENHAHN,
OLIVER
GRANERT,
AND
GERALD
SOMMER
373
33.1
INTRODUCTION
.
373
33.2
POSE
ESTIMATION
IN
CONFORMAL
GEOMETRIC
ALGEBRA
.
374
33.2.1
THE
SCENARIO
OF
POSE
ESTIMATION
.
374
33.2.2
INTRODUCTION
TO
CONFORMAL
GEOMETRIC
ALGEBRA
.
375
XXII
CONTENTS
33.2.3
KINEMATIC
CONSTRAINTS
IN
CONFORMAL
GEOMETRIC
ALGEBRA
.
376
33.3
POSE
ESTIMATION
OF
KINEMATIC
CHAINS
.
377
33.3.1
KINEMATIC
CHAINS
IN
CONFORMAL
GEOMETRIC
ALGEBRA
.
377
33.3.2
CONSTRAINT
EQUATIONS
FOR
KINEMATIC
CHAINS
.
378
33.4
EXPERIMENTS
.
379
33.5
DISCUSSION
.
381
34
STABILIZATION
OF
3D
POSE
ESTIMATION
W.
NEDDERMEYER,
M.
SCHNELL,
W.
WINKLER,
AND
A.
LILIENTHAL
385
34.1
INTRODUCTION
.
385
34.2
CAMERA
MODEL
.
385
34.3
DETERMINATION
OF
POSITION
.
387
34.3.1
DETERMINATION
OF
POSITION
BY
THE
BUNDLE
ADJUSTMENT
METHOD
.
387
34.3.2
REFERENCE
MEASUREMENT
.
388
34.4
STABILIZATION
.
389
34.4.1
STABILIZATION
OF
THE
OPERATING
POINT
BY
VARIATION
OF
CAMERA
POSITION
.
389
34.4.2
STABILIZATION
OF
THE
OPERATING
POINT
BY
VARIATION
OF
THE
OBJECT
PARAMETERS
.
391
34.4.3
EXAMPLE
.
392
34.5
CONCLUDING
REMARKS
.
392
35
INFERRING
DYNAMICAL
INFORMATION
FROM
3D
POSITION
DATA
USING
GEOMETRIC
ALGEBRA
HINIDUMA
UDUGAMA
GAMAGE
SAHAN
SAJEEWA
AND
JOAN
LASENBY
395
35.1
INTRODUCTION
.
395
35.2
SOME
BASIC
FORMULATIONS
.
396
35.2.1
ANGULAR
VELOCITY
.
396
35.2.2
LINEAR
VELOCITY,
ACCELERATION
AND
INERTIAL
FORCE
.
.
.
397
35.2.3
ANGULAR
MOMENTUM,
INERTIA
TENSOR
AND
INERTIAL
TORQUE
.
397
35.3
CALCULATIONS
IN
TERMS
OF
ROTATIONAL
BIVECTORS
.
398
35.4
ALGORITHM
FOR
INVERSE
DYNAMICS
.
399
35.5
DYNAMICAL
EQUILIBRIUM
IN
THE
MODEL
.
401
35.6
INVERSE
DYNAMICS
FROM
MOTION
CAPTURE
DATA
.
402
35.7
REAL
WORLD
APPLICATIONS
AND
RESULTS
.
403
35.8
CONCLUSIONS
AND
FUTURE
WORK
.
405
36
CLIFFORD
ALGEBRA
SPACE
SINGULARITIES
OF
INLINE
PLANAR
PLATFORMS
MICHAEL
A.
BASWELL,
RAFAL
ABLAMOWICZ,
AND
JOE
N.
ANDERSON
407
36.1
INTRODUCTION
.
407
36.1.1
VGT
SINGULARITIES
.
408
CONTENTS
XXIII
36.1.2
PAPER
ORGANIZATION
.
409
36.2
SINGULARITY
TYPES
AND
CLIFFORD
ALGEBRAS
.
409
36.2.1
TYPE
I,
TYPE
II,
AND
TYPE
III
SINGULARITIES
.
409
36.2.2
THE
CLIFFORD
ALGEBRA
C
+
(P
3
)
.
410
36.2.3
C
+
(P
3
)
COMPONENTS
FROM
THE
SCREW
PARAMETERIZATION
.
411
36.3
SINGULARITY
SURFACES
OF
SINGLE
STAGE
PLANAR
PLATFORMS
.
413
36.3.1
THE
HOMOGENEOUS
TRANSFORMATIONS
.
413
36.3.2
THE
CONSTRAINT
MANIFOLD
.
413
36.3.3
THE
QUATERNIONIC
JACOBIAN
.
415
36.3.4
SINGULARITY
SETS
OF
PLANAR
PLATFORMS
.
416
36.4
SINGULARITY
SURFACES
OF
TWO
STACKED
PLANAR
PLATFORMS
.
.
.
418
36.4.1
THE
JACOBIAN
OF
TWO
STACKED
GENERAL
INLINE
PLANAR
PLATFORMS
.
419
36.5
CONCLUSIONS
AND
RECOMMENDATIONS
.
420
IV
SIGNAL
PROCESSING
AND
OTHER
APPLICATIONS
423
37
FAST
QUANTUM
FOURIER
-
HEISENBERG
-
WEYL
TRANSFORMS
V.
LABUNETS,
E.
RUNDBLAD,
AND
J.
ASTOLA
425
37.1
INTRODUCTION
.
425
37.2
HEISENBERG-WEYL
GROUPS
.
427
37.3
QUANTUM
FOURIER-HEISENBERG-WEYL
TRANSFORM
.
429
38
THE
STRUCTURE
MULTIVECTOR
MICHAEL
FELSBERG
AND
GERALD
SOMMER
437
38.1
INTRODUCTION
AND
MOTIVATION
.
437
38.2
MATHEMATICAL
FUNDAMENTALS
.
440
38.3
THE
APPROACH
FOR
ILD
SYMMETRIES
.
442
38.4
THE
APPROACH
FOR
I2D
SYMMETRIES
.
443
38.5
THE
STRUCTURE
MULTIVECTOR
.
444
38.6
CONCLUSION
.
446
39
THE
APPLICATION
OF
CLIFFORD
ALGEBRA
TO
CALCULATIONS
OF
MULTICOMPONENT
CHEMICAL
COMPOSITION
JOHN
P.
FLETCHER
449
39.1
INTRODUCTION
.
449
39.2
PROJECTION
.
450
39.3
THE
V
PRODUCT
.
451
39.4
MULTICOMPONENT
CHEMICAL
COMPOSITION
.
451
39.4.1
PROJECTION
MODEL
.
451
39.4.2
CONVERSION
BETWEEN
MASS
AND
MOLAR
BASIS
.
453
39.4.3
VAPOUR-LIQUID
EQUILIBRIUM
.
454
39.4.4
FLASH
CALCULATION
.
455
XXIV
CONTENTS
39.4.5
CHEMICAL
REACTION
.
455
39.4.6
VOLUME
AND
THERMODYNAMIC
FUNCTIONS
.
456
39.5
VEE
PRODUCT
MODEL
.
456
39.6
GENERALISATION
.
458
39.7
CONCLUSIONS
.
458
40
AN
ALGORITHM
TO
SOLVE
THE
INVERSE
IFS-PROBLEM
ERWIN
HOCEVAR
459
40.1
INTRODUCTION
.
459
40.1.1
PROBLEM
SPECIFICATION
-
OBJECTS
TO
BE
ENCODED
.
.
459
40.2
ALGORITHM
TO
CALCULATE
THE
IFS-CODES
OF
AN
IMAGE
.
460
40.2.1
CALCULATING
BOUNDARIES
OF
AN
OBJECT
.
460
40.2.2
CALCULATING
ORBITS
DEFINING
A
NOT
MINIMAL
IFS
.
.
.
461
40.2.3
CALCULATING
A
MINIMAL
IFS
.
462
40.2.4
CALCULATING
IFS-CODES
FOR
THE
INNER
IMAGE
PARTS
.
.
462
40.3
CONCLUSION
.
463
40.3.1
SUMMARY
.
463
40.3.2
FUTURE
WORK
.
464
41
FAST
QUANTUM
N-D
FOURIER
AND
RADON
TRANSFORMS
V.
LABUNETS,
E.
RUNDBLAD,
AND
J.
ASTOLA
469
41.1
INTRODUCTION
.
469
41.2
DISCRETE
RADON
AND
/C-TRANSFORMS
.
469
41.3
FAST
CLASSICAL
RADON
AND
AS-TRANSFORMS
.
471
41.3.1
FAST
2D
RADON
AND
V-TRANSFORMS
.
471
41.3.2
FAST
ND
AC-TRANSFORM
ON
ZP
-1
X
MZ
P
.
473
41.3.3
FAST
N.D
/C-TRANSFORM
ON
THE
Z
"
YY
1
X
MZ
P
YY
.
474
41.4
FAST
QUANTUM
FOURIER,
RADON
AND
V-TRANSFORMS
.
475 |
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| genre | (DE-588)1071861417 Konferenzschrift 2001 Cambridge gnd-content |
| genre_facet | Konferenzschrift 2001 Cambridge |
| id | DE-604.BV014363390 |
| illustrated | Illustrated |
| indexdate | 2024-08-24T00:30:32Z |
| institution | BVB |
| isbn | 3764342676 0817642676 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009843855 |
| oclc_num | 49276467 |
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| owner | DE-739 DE-703 DE-355 DE-BY-UBR |
| owner_facet | DE-739 DE-703 DE-355 DE-BY-UBR |
| physical | XXIV, 478 S. Ill., graph. Darst. : 24 cm |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Birkhäuser |
| record_format | marc |
| spelling | Applications of geometric algebra in computer science and engineering Leo Dorst ... (ed.) Boston ; Basel ; Berlin Birkhäuser 2002 XXIV, 478 S. Ill., graph. Darst. : 24 cm txt rdacontent n rdamedia nc rdacarrier Informatik Mathematik Computer science Mathematics Engineering mathematics Geometry, Algebraic Geometrische Algebra (DE-588)4156707-9 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 2001 Cambridge gnd-content Geometrische Algebra (DE-588)4156707-9 s DE-604 Dorst, Leo 1958- Sonstige (DE-588)114432058 oth DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009843855&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Applications of geometric algebra in computer science and engineering Informatik Mathematik Computer science Mathematics Engineering mathematics Geometry, Algebraic Geometrische Algebra (DE-588)4156707-9 gnd |
| subject_GND | (DE-588)4156707-9 (DE-588)1071861417 |
| title | Applications of geometric algebra in computer science and engineering |
| title_auth | Applications of geometric algebra in computer science and engineering |
| title_exact_search | Applications of geometric algebra in computer science and engineering |
| title_full | Applications of geometric algebra in computer science and engineering Leo Dorst ... (ed.) |
| title_fullStr | Applications of geometric algebra in computer science and engineering Leo Dorst ... (ed.) |
| title_full_unstemmed | Applications of geometric algebra in computer science and engineering Leo Dorst ... (ed.) |
| title_short | Applications of geometric algebra in computer science and engineering |
| title_sort | applications of geometric algebra in computer science and engineering |
| topic | Informatik Mathematik Computer science Mathematics Engineering mathematics Geometry, Algebraic Geometrische Algebra (DE-588)4156707-9 gnd |
| topic_facet | Informatik Mathematik Computer science Mathematics Engineering mathematics Geometry, Algebraic Geometrische Algebra Konferenzschrift 2001 Cambridge |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009843855&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT dorstleo applicationsofgeometricalgebraincomputerscienceandengineering |