Geometric Algebra with applications in science and engineering:
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Boston [u.a.]
Birkhäuser
2001
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXVI, 592 S. Ill., graph. Darst. |
| ISBN: | 0817641998 3764341998 |
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| 016 | 7 | |a 961996277 |2 DE-101 | |
| 020 | |a 0817641998 |9 0-8176-4199-8 | ||
| 020 | |a 3764341998 |9 3-7643-4199-8 | ||
| 035 | |a (OCoLC)45052642 | ||
| 035 | |a (DE-599)BVBBV013862939 | ||
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| 084 | |a SK 240 |0 (DE-625)143226: |2 rvk | ||
| 245 | 1 | 0 | |a Geometric Algebra with applications in science and engineering |c Eduardo Bayro Corrochano ... [Eds.] |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2001 | |
| 300 | |a XXVI, 592 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 7 | |a Geometrische algebra |2 gtt | |
| 650 | 4 | |a Geometry, Algebraic | |
| 650 | 0 | 7 | |a Geometrische Algebra |0 (DE-588)4156707-9 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Geometrische Algebra |0 (DE-588)4156707-9 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Bayro Corrochano, Eduardo |e Sonstige |0 (DE-588)123058538 |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=009482120&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-009482120 | |
Datensatz im Suchindex
| _version_ | 1807593423938519040 |
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| adam_text |
CONTENTS
PREFACE
XVII
CONTRIBUTORS
XXI
I
ADVANCES
IN
GEOMETRIC
ALGEBRA
1
1
OLD
WINE
IN
NEW
BOTTLES:
A
NEW
ALGEBRAIC
FRAMEWORK
FOR
COMPUTATIONAL
GEOMETRY
DAVID
HESTENES
3
1.1
INTRODUCTION
.
3
1.2
MINKOWSKI
ALGEBRA
.
3
1.3
CONFORMAL
SPLIT
.
5
1.4
MODELS
OF
EUCLIDEAN
SPACE
.
5
1.5
LINES
AND
PLANES
.
7
1.6
SPHERES
AND
HYPERPLANES
.
9
1.7
CONFORMAL
AND
EUCLIDEAN
GROUPS
.
12
1.8
SCREW
MECHANICS
.
15
1.9
CONCLUSIONS
.
16
2
UNIVERSAL
GEOMETRIC
ALGEBRA
GARRET
SOBCZYK
18
2.1
INTRODUCTION
.
18
2.2
THE
UNIVERSAL
GEOMETRIC
ALGEBRA
.
19
2.2.1
THE
STANDARD
BASIS
.
22
2.3
MATRICES
OF
GEOMETRIC
NUMBERS
.
22
2.3.1
RECIPROCAL
BASIS
.
24
2.3.2
GENERALIZED
INVERSE
OF
A
MATRIX
.
25
2.3.3
THE
MEET
AND
JOINT
OPERATIONS
.
26
2.3.4
HERMITIAN
INNER
PRODUCT
.
28
2.4
LINEAR
TRANSFORMATIONS
.
29
2.4.1
SPECTRAL
DECOMPOSITION
OF
A
LINEAR
OPERATOR
.
29
2.4.2
PRINCIPAL
CORRELATION
.
31
2.4.3
THE
BIVECTOR
OF
A
LINEAR
OPERATOR
.
32
2.4.4
SPINOR
REPRESENTATION
.
33
2.5
PSEUDO-EUCLIDEAN
GEOMETRIES
.
34
2.6
AFFINE
AND
PROJECTIVE
GEOMETRIES
.
35
2.6.1
PSEUDO-AFFINE
GEOMETRIES
.
36
VI
CONTENTS
2.6.2
DESARGUES
'
THEOREM
.
37
2.6.3
SIMPSON
'
S
THEOREM
FOR
THE
CIRCLE
.
38
2.7
CONFORMAL
TRANSFORMATIONS
.
40
3
REALIZATIONS
OF
THE
CONFORMAL
GROUP
JOSE
MARIA
POZO
AND
GARRET
SOBCZYK
42
3.1
INTRODUCTION
.
42
3.2
PROJECTIVE
GEOMETRY
.
43
3.3
THE
CONFORMAL
REPRESENTANT
AND
STEREOGRAPHIC
PROJECTION
.
44
3.4
CONFORMAL
TRANSFORMATIONS
AND
ISOMETRIES
.
48
3.5
ISOMETRIES
IN
IN
Q
.
49
3.6
COMPACTIFICATION
.
53
3.7
MOBIUS
TRANSFORMATIONS
.
55
4
HYPERBOLIC
GEOMETRY
HONGBO
LI
61
4.1
INTRODUCTION
.
61
4.2
HYPERBOLIC
PLANE
GEOMETRY
WITH
CLIFFORD
ALGEBRA
.
62
4.2.1
GENERALIZED
TRIANGLES
.
63
4.2.2
THE
AREA
AND
PERIMETER
OF
A
CONVEX
N-POLYGON
.
66
4.3
HYPERBOLIC
CONFORMAL
GEOMETRY
WITH
CLIFFORD
ALGEBRA
.
67
4.3.1
DOUBLE-HYPERBOLIC
SPACE
.
68
4.3.2
THE
HOMOGENEOUS
MODEL
OF
A
DOUBLE-HYPERBOLIC
SPACE
.
69
4.3.3
BUNCHES
OF
TOTAL
SPHERES
.
71
4.3.4
CONFORMAL
TRANSFORMATIONS
.
74
4.4
A
UNIVERSAL
MODEL
FOR
THE
CONFORMAL
GEOMETRIES
OF
THE
EUCLIDEAN,
SPHERICAL,
AND
DOUBLE-HYPERBOLIC
SPACES
.
77
4.4.1
THE
HOMOGENEOUS
MODEL
OF
THE
EUCLIDEAN
SPACE
.
78
4.4.2
THE
HOMOGENEOUS
MODEL
OF
THE
SPHERICAL
SPACE
.
79
4.4.3
A
UNIVERSAL
MODEL
FOR
THREE
GEOMETRIES
.
81
4.5
CONCLUSION
.
85
II
THEOREM
PROVING
87
5
GEOMETRIC
REASONING
WITH
GEOMETRIC
ALGEBRA
DONGMING
WANG
89
5.1
INTRODUCTION
.
89
5.2
CLIFFORD
ALGEBRA
FOR
EUCLIDEAN
GEOMETRY
.
90
5.3
GEOMETRIC
THEOREM
PROVING
.
93
5.3.1
DERIVING
REPRESENTATIONS
OF
GEOMETRIC
OBJECTS
.
93
5.3.2
EXAMPLES
OF
THEOREM
PROVING
.
95
CONTENTS
VII
5.3.3
APPROACHES
TO
GEOMETRIC
REASONING
.
99
5.4
PROVING
IDENTITIES
IN
CLIFFORD
ALGEBRA
.
100
5.4.1
INTRODUCTION
BY
EXAMPLES
.
100
5.4.2
PRINCIPLES
AND
TECHNIQUES
.
106
6
AUTOMATED
THEOREM
PROVING
HONGBO
LI
110
6.1
INTRODUCTION
.
110
6.2
A
GENERAL
FRAMEWORK
FOR
CLIFFORD
ALGEBRA
AND
WU
'
S
METHOD
.
ILL
6.3
AUTOMATED
THEOREM
PROVING
IN
EUCLIDEAN
GEOMETRY
AND
OTHER
CLASSICAL
GEOMETRIES
.
114
6.4
AUTOMATED
THEOREM
PROVING
IN
DIFFERENTIAL
GEOMETRY
.
116
6.5
CONCLUSION
.
119
III
COMPUTER
VISION
121
7
THE
GEOMETRY
ALGEBRA
OF
COMPUTER
VISION
EDUARDO
BAYRO
CORROCHANO
AND
JOAN
LASENBY
123
7.1
INTRODUCTION
.
123
7.2
THE
GEOMETRIC
ALGEBRAS
OF
3-D
AND
4-D
SPACES
.
124
7.2.1
3-D
SPACE
AND
THE
2-D
IMAGE
PLANE
.
124
7.2.2
THE
GEOMETRIC
ALGEBRA
OF
3-D
EUCLIDEAN
SPACE
.
126
7.2.3
A
4-D
GEOMETRIC
ALGEBRA
FOR
PROJECTIVE
SPACE
.
127
7.2.4
PROJECTIVE
TRANSFORMATIONS
.
127
7.2.5
THE
PROJECTIVE
SPLIT
.
129
7.3
THE
ALGEBRA
OF
INCIDENCE
.
131
7.3.1
THE
BRACKET
.
YY
.
131
7.3.2
THE
DUALITY
PRINCIPLE
AND
THE
MEET
AND
JOIN
OPERATIONS
.
132
7.3.3
LINEAR
ALGEBRA
.
134
7.4
ALGEBRA
IN
PROJECTIVE
SPACE
.
135
7.4.1
INTERSECTION
OF
A
LINE
AND
A
PLANE
.
135
7.4.2
INTERSECTION
OF
TWO
PLANES
.
136
7.4.3
INTERSECTION
OF
TWO
LINES
.
137
7.4.4
IMPLEMENTATION
OF
THE
ALGEBRA
.
137
7.5
VISUAL
GEOMETRY
OF
N
UNCALIBRATED
CAMERAS
.
137
7.5.1
GEOMETRY
OF
ONE
VIEW
.
138
7.5.2
GEOMETRY
OF
TWO
VIEWS
.
141
7.5.3
GEOMETRY
OF
THREE
VIEWS
.
144
7.5.4
GEOMETRY
OF
N-VIEWS
.
145
7.6
CONCLUSIONS
.
146
VIII
CONTENTS
8
USING
GEOMETRIC
ALGEBRA
FOR
OPTICAL
MOTION
CAPTURE
JOAN
LASENBY
AND
ADAM
STEVENSON
147
8.1
INTRODUCTION
.
147
8.2
EXTERNAL
AND
INTERNAL
CALIBRATION
.
149
8.2.1
EXTERNAL
CALIBRATION
.
149
8.2.2
INTERNAL
CALIBRATION
.
149
8.3
ESTIMATING
THE
EXTERNAL
PARAMETERS
.
151
8.3.1
DIFFERENTIATION
W.R.T.
TK
.
153
8.3.2
DIFFERENTIATION
W.R.T.
.
154
8.3.3
DIFFERENTIATION
W.R.T.
THE
X
PQ
^
.
156
8.3.4
DIFFERENTIATION
W.R.T.
THE
X^
.
156
8.3.5
REFINING
THE
ESTIMATES
OF
TJ
AND
.
156
8.3.6
OPTIMAL
RECONSTRUCTION
FROM
CALIBRATED
DATA
.
157
8.3.7
AN
INITIAL
ESTIMATE
FOR
THE
TRANSLATIONS
.
158
8.3.8
THE
ITERATIVE
CALIBRATION
SCHEME
.
159
8.4
EXAMPLES
AND
RESULTS
.
160
8.5
EXTENDING
TO
INCLUDE
INTERNAL
CALIBRATION
.
164
8.6
CONCLUSIONS
.
167
9
BAYESIAN
INFERENCE
AND
GEOMETRIC
ALGEBRA:
AN
APPLICATION
TO
CAMERA
LOCALIZATION
CHRIS
DORAN
170
9.1
INTRODUCTION
.
170
9.2
GEOMETRIC
ALGEBRA
IN
THREE
DIMENSIONS
.
171
9.3
ROTORS
AND
ROTATIONS
.
172
9.3.1
THE
GROUP
MANIFOLD
.
173
9.3.2
EXTRAPOLATING
BETWEEN
ROTATIONS
.
174
9.4
ROTOR
CALCULUS
.
175
9.5
COMPUTER
VISION
.
178
9.5.1
KNOWN
RANGE
DATA
.
178
9.5.2
SOLUTION
.
181
9.5.3
ADDING
MORE
CAMERAS
.
182
9.6
UNKNOWN
RANGE
DATA
.
184
9.7
EXTENSION
TO
THREE
CAMERAS
.
188
9.8
CONCLUSIONS
.
189
10
PROJECTIVE
RECONSTRUCTION
OF
SHAPE
AND
MOTION
USING
IN
VARIANT
THEORY
EDUARDO
BAYRO
CORROCHANO
AND
VLADIMIR
BANARER
190
10.1
INTRODUCTION
.
190
10.2
3-D
PROJECTIVE
INVARIANTS
FROM
MULTIPLE
VIEWS
.
191
10.2.1
GENERATION
OF
GEOMETRIC
PROJECTIVE
INVARIANTS
.
191
10.2.2
PROJECTIVE
INVARIANTS
USING
TWO
VIEWS
.
194
10.2.3
PROJECTIVE
INVARIANT
OF
POINTS
USING
THREE
VIEWS
.
.
.
197
10.2.4
COMPARISON
OF
THE
PROJECTIVE
INVARIANTS
.
198
10.3
PROJECTIVE
DEPTH
.
200
CONTENTS
IX
10.4
SHAPE
AND
MOTION
.
202
10.4.1
THE
JOIN
IMAGE
.
203
10.4.2
THE
SVD
METHOD
.
204
10.4.3
COMPLETION
OF
THE
3-D
SHAPE
USING
GEOMETRIC
INVARIANTS
.
206
10.5
CONCLUSIONS
.
207
IV
ROBOTICS
209
11
ROBOT
KINEMATICS
AND
FLAGS
J.M.
SELIG
211
11.1
INTRODUCTION
.
211
11.2
THE
CLIFFORD
ALGEBRA
.
211
11.2.1
THE
GROUP
OF
RIGID
BODY
MOTIONS
.
212
11.2.2
POINTS,
LINES
AND
PLANES
.
213
11.2.3
SOME
RELATIONS
.
215
11.3
FLAGS
.
216
11.3.1
POINTED
LINES
.
217
11.3.2
POINTED
PLANES
.
218
11.3.3
LINED
PLANES
.
219
11.3.4
COMPLETE
FLAGS
.
220
11.4
ROBOTS
.
221
11.4.1
KINEMATICS
.
221
11.4.2
PIEPER
'
S
THEOREM
.
222
11.4.3
EXAMPLE
-
THE
MA2000
.
225
11.4.4
EXAMPLE
-
THE
INTELLEDEX
660
.
229
11.5
CONCLUDING
REMARKS
.
233
12
THE
CLIFFORD
ALGEBRA
AND
THE
OPTIMIZATION
OF
ROBOT
DE
SIGN
SHAWN
G.
AHLERS
AND
JOHN
MICHAEL
MCCARTHY
235
12.1
INTRODUCTION
.
235
12.2
LITERATURE
REVIEW
.
235
12.3
OVERVIEW
OF
THE
DESIGN
ALGORITHM
.
236
12.4
DOUBLE
QUATERNIONS
.
237
12.4.1
HOMOGENEOUS
TRANSFORMS
.
.
237
12.4.2
THE
CLIFFORD
ALGEBRA
ON
E
4
.
238
12.4.3
HOMOGENEOUS
TRANSFORMATIONS
AS
A
ROTATIONS
IN
E
4
.
239
12.4.4
DOUBLE
QUATERNION
FOR
A
SPATIAL
DISPLACEMENT
.
240
12.4.5
SPATIAL
DISPLACEMENT
FROM
A
DOUBLE
QUATERNION
.
.
.
241
12.5
THE
TASK
TRAJECTORY
.
242
12.5.1
THE
DECASTELJAU
ALGORITHM
.
242
12.5.2
BEZIER
INTERPOLATION
.
243
X
CONTENTS
12.5.3
EXAMPLE
OF
DOUBLE
QUATERNION
INTERPOLATION
.
247
12.6
THE
DESIGN
OF
THE
TS
ROBOT
.
248
12.7
THE
OPTIMUM
TS
ROBOT
.
249
12.7.1
THE
OPTIMUM
TS
ROBOT
AND
THE
TASK
TRAJECTORY
.
.
.
250
12.8
CONCLUSION
.
251
13
APPLICATIONS
OF
LIE
ALGEBRAS
AND
THE
ALGEBRA
OF
INCIDENCE
EDUARDO
BAYRO
CORROCHANO
AND
GARRET
SOBCZYK
252
13.1
INTRODUCTION
.
252
13.2
THE
GENERAL
LINEAR
GROUP
.
253
13.2.1
THE
ORTHOGONAL
GROUPS
.
255
13.2.2
THE
LIE
GROUP
AND
LIE
ALGEBRA
OF
THE
AFFINE
PLANE
.
257
13.3
ALGEBRA
OF
INCIDENCE
.
262
13.3.1
INCIDENCE
RELATIONS
IN
THE
AFFINE
N-PLANE
.
263
13.3.2
INCIDENCE
RELATIONS
IN
THE
AFFINE
3-PLANE
.
264
13.3.3
GEOMETRIC
CONSTRAINTS
AS
INDICATORS
.
266
13.4
RIGID
MOTION
IN
THE
AFFINE
PLANE
.
267
13.5
APPLICATION
TO
ROBOTICS
.
268
13.5.1
INVERSE
KINEMATIC
COMPUTING
.
268
13.5.2
ROBOT
MANIPULATION
GUIDANCE
.
271
13.5.3
CHECKING
FOR
A
CRITICAL
CONFIGURATION
.
271
13.6
APPLICATION
II:
IMAGE
ANALYSIS
.
273
13.6.1
THE
DESIGN
OF
AN
IMAGE
FILTER
.
273
13.6.2
RECOGNITION
OF
HAND
GESTURES
.
274
13.6.3
THE
MEET
FILTER
.
275
13.7
CONCLUSION
.
276
V
QUANTUM
AND
NEURAL
COMPUTING,
AND
WAVELETS
279
14
GEOMETRIC
ALGEBRA
IN
QUANTUM
INFORMATION
PROCESSING
BY
NUCLEAR
MAGNETIC
RESONANCE
TIMOTHY
F.
HAVEL,
DAVID
G.
CORY,
SHYAMAL
S.
SOMAROO,
AND
CHING
HUA
TSENG
281
14.1
INTRODUCTION
.
281
14.2
MULTIPARTICLE
GEOMETRIC
ALGEBRA
.
283
14.3
ALGORITHMS
FOR
QUANTUM
COMPUTERS
.
285
14.4
NMR
AND
THE
PRODUCT
OPERATOR
FORMALISM
.
288
14.5
QUANTUM
COMPUTING
BY
LIQUID-STATE
NMR
.
292
14.6
STATES
AND
GATES
BY
NMR
.
297
14.7
QUANTUM
SIMULATION
BY
NMR
.
301
14.8
REMARKS
ON
FOUNDATIONAL
ISSUES
.
306
CONTENTS
XI
15
GEOMETRIC
FEEDFORWARD
NEURAL
NETWORKS
AND
SUPPORT
MUL
TIVECTOR
MACHINES
EDUARDO
BAYRO
CORROCHANO
AND
REFUGIO
VALLEJO
309
15.1
INTRODUCTION
.
309
15.2
REAL
VALUED
NEURAL
NETWORKS
.
310
15.3
COMPLEX
MLP
AND
QUATERNIONIC
MLP
.
311
15.4
GEOMETRIC
ALGEBRA
NEURAL
NETWORKS
.
312
15.4.1
THE
ACTIVATION
FUNCTION
.
312
15.4.2
THE
GEOMETRIC
NEURON
.
313
15.4.3
FEEDFORWARD
GEOMETRIC
NEURAL
NETWORKS
.
314
15.5
LEARNING
RULE
.
316
15.5.1
MULTI-DIMENSIONAL
BACK-PROPAGATION
TRAINING
RULE
.
.
316
15.6
EXPERIMENTS
USING
GEOMETRIC
FEEDFORWARD
NEURAL
NETWORKS
.
317
15.7
SUPPORT
VECTOR
MACHINES
IN
GEOMETRIC
ALGEBRA
.
319
15.7.1
SUPPORT
VECTOR
MACHINES
.
319
15.7.2
SUPPORT
MULTIVECTOR
MACHINES
.
320
15.8
EXPERIMENTAL
ANALYSIS
OF
SUPPORT
MULTIVECTOR
MACHINES
.
.
321
15.8.1
FINDING
SUPPORT
MULTIVECTORS
.
321
15.8.2
ESTIMATION
OF
3D
RIGID
MOTION
.
322
15.9
CONCLUSIONS
.
324
16
IMAGE
ANALYSIS
USING
QUATERNION
WAVELETS
LEONARDO
TRAVERSONI
326
16.1
INTRODUCTION
.
326
16.1.1
WAVELETS
.
326
16.1.2
MULTIRRESOLUTION
ANALYSIS
.
327
16.1.3
CARDINAL
SPLINES
.
328
16.1.4
DECOMPOSITION
AND
RECONSTRUCTION
.
328
16.1.5
HILBERT
SPACES
OF
QUATERNIONIC
VALUED
FUNCTIONS
.
.
.
329
16.1.6
MODULES
OVER
QUATERNIONS
.
331
16.1.7
HILBERT
QUATERNION
MODULES
.
332
16.1.8
HILBERT
MODULES
WITH
REPRODUCING
KERNEL
.
332
16.1.9
KERNEL
.
332
16.2
THE
STATIC
APPROACH
.
336
16.3
CLIFFORD
MULTIRESOLUTION
ANALYSES
OF
L
2
(LLR
M
)
C(
N
)
.
337
16.4
HAAR
QUATERNIONIC
WAVELETS
.
338
16.4.1
DECOMPOSITION
AND
RECONSTRUCTION
FOR
QUATERNION
WAVELETS
.
339
16.4.2
A
BIOMEDICAL
APPLICATION
.
339
16.5
A
DYNAMIC
INTERPRETATION
.
343
16.6
GLOBAL
INTERPOLATION
.
344
16.7
DEALING
WITH
TRAJECTORIES
.
344
16.8
CONCLUSIONS
.
345
XII
CONTENTS
VI
APPLICATIONS
TO
ENGINEERING
AND
PHYSICS
347
17
OBJECTS
IN
CONTACT:
BOUNDARY
COLLISIONS
EIS
GEOMETRIC
WAVE
PROPAGATION
LEO
DORST
349
17.1
INTRODUCTION
.
349
17.1.1
TOWARDS
A
'
SYSTEMS
THEORY
'
OF
COLLISION
.
349
17.1.2
COLLISION
IS
LIKE
WAVE
PROPAGATION
.
349
17.1.3
RELATED
PROBLEMS
.
351
17.2
BOUNDARY
GEOMETRY
.
352
17.2.1
THE
ORIENTED
TANGENT
SPACE
.
352
17.2.2
DIFFERENTIAL
GEOMETRY
OF
THE
BOUNDARY
.
353
17.3
THE
BOUNDARY
AS
A
GEOMETRIC
OBJECT
.
354
17.3.1
EMBEDDING
IN
CF
M
+I,I
.
355
17.3.2
BOUNDARIES
REPRESENTED
IN
Q{EL
M
)
.
356
17.3.3
EXAMPLE:
A
SPHERICAL
BOUNDARY
.
359
17.3.4
BOUNDARIES
AS
DIRECTION-DEPENDENT
ROTORS
.
361
17.3.5
THE
EFFECT
OF
EUCLIDEAN
TRANSFORMATIONS
.
362
17.4
WAVE
PROPAGATION
OF
BOUNDARIES
.
363
17.4.1
DEFINITION
OF
PROPAGATION
.
363
17.4.2
PROPAGATION
IN
THE
EMBEDDED
REPRESENTATIONS
.
364
17.4.3
A
SYSTEMS
THEORY
OF
WAVE
PROPAGATION
AND
COLLISION
.
365
17.4.4
MATCHING
TANGENTS
.
366
17.4.5
EXAMPLES
OF
PROPAGATION
.
366
17.4.6
ANALYSIS
OF
PROPAGATION
.
368
17.5
CONCLUSIONS
.
369
18
MODERN
GEOMETRIC
CALCULATIONS
IN
CRYSTALLOGRAPHY
G.
ARAGON,
J.L.
ARAGON,
F.
DAVILA,
A.
GOMEZ
AND
M.A.
RODRIGUEZ371
18.1
INTRODUCTION
.
371
18.2
QUASICRYSTALS
.
372
18.3
THE
MORPHOLOGY
OF
ICOSAHEDRAL
QUASICRYSTALS
.
375
18.4
COINCIDENCE
SITE
LATTICE
THEORY
.
381
18.4.1
BASICS
.
381
18.4.2
GEOMETRIC
ALGEBRA
APPROACH
.
383
18.5
CONCLUSIONS
.
386
19
QUATERNION
OPTIMIZATION
PROBLEMS
IN
ENGINEERING
LJUDMILA
MEISTER
387
19.1
INTRODUCTION
.
387
19.2
PROPERTIES
OF
QUATERNIONS
.
388
19.2.1
NOTATIONS
AND
DEFINITIONS
.
388
19.2.2
QUATERNIONS
AND
VECTORS
.
389
19.2.3
QUATERNION
DESCRIPTION
OF
ROTATIONS
.
391
CONTENTS
XIII
19.3
EXTREMAL
PROBLEMS
FOR
QUATERNIONS
.
393
19.3.1
DIFFERENTIATION
WITH
RESPECT
TO
QUATERNIONS
.
393
19.3.2
MINIMIZATION
OF
LOSS
FUNCTIONS
.
394
19.3.3
CONDITIONAL
EXTREMUM
PROBLEMS
.
394
19.3.4
THE
LEAST-SQUARES
METHOD
FOR
QUATERNIONS
.
395
19.4
DETERMINATION
OF
ROTATIONS
.
396
19.4.1
UNKNOWN
ROTATION
OF
A
VECTOR
.
396
19.4.2
ROTATION
OF
SEVERAL
VECTORS
.
401
19.5
THE
MAIN
PROBLEM
OF
ORIENTATION
.
401
19.5.1
ORIENTATION
BASED
ON
FREE
VECTORS
.
402
19.5.2
ROTO-TRANSLATION
PROBLEM
.
403
19.5.3
PHOTO
EXTERIOR
ORIENTATION
PROBLEM
.
403
19.5.4
ORIENTATION
BASED
ON
COPLANAR
VECTORS
.
405
19.5.5
RELATIVE
ORIENTATION
OF
A
STEREO-PAIR
.
405
19.6
OPTIMAL
FILTERING
AND
PREDICTION
.
407
19.6.1
RANDOM
QUATERNIONS
.
407
19.6.2
OPTIMAL
FILTERING
AND
PREDICTION
FOR
SINGLE-STAGE
ROTATIONS
.
408
19.6.3
OPTIMAL
FILTERING
AND
PREDICTION
FOR
MULTI-STAGE
ROTATIONS
.
410
19.6.4
THE
LAW
OF
LARGE
NUMBERS
.
411
19.7
SUMMARY
.
411
20
CLIFFORD
ALGEBRAS
IN
ELECTRICAL
ENGINEERING
WILLIAM
E.
BAYLIS
413
20.1
INTRODUCTION
.
413
20.2
STRUCTURE
OF
D?3
.
415
20.2.1
CLIFFORD
DUAL
AND
CONJUGATIONS
.
415
20.3
PARAVECTOR
MODEL
OF
SPACETIME
.
416
20.3.1
SPACETIME
PLANES:
BIPARAVECTORS
.
417
20.3.2
LORENTZ
TRANSFORMATIONS
AND
COVARIANCE
.
418
20.4
USING
RELATIVITY
AT
LOW
SPEEDS
.
419
20.4.1
MAXWELL
AND
CONTINUITY
EQUATIONS
.
419
20.4.2
CONDUCTING
SCREEN
.
420
20.4.3
LORENTZ
FORCE
.
423
20.4.4
WAVE
GUIDES
.
424
20.5
RELATIVITY
AT
HIGH
SPEEDS
.
424
20.5.1
MOTION
OF
CHARGES
IN
FIELDS
.
424
20.5.2
VIRTUAL
PHOTON
SHEETS
.
427
20.6
CONCLUSIONS
.
429
21
APPLICATIONS
OF
GEOMETRIC
ALGEBRA
IN
PHYSICS
AND
LINKS
WITH
ENGINEERING
ANTHONY
LASENBY
AND
JOAN
LASENBY
430
21.1
INTRODUCTION
.
430
XIV
CONTENTS
21.2
THE
SPACETIME
ALGEBRA
.
431
21.2.1
THE
SPACETIME
SPLIT,
SPECIAL
RELATIVITY,
AND
ELECTROMAGNETISM
.
431
21.3
QUANTUM
MECHANICS
.
435
21.4
GRAVITY
AS
A
GAUGE
THEORY
.
438
21.4.1
SOME
APPLICATIONS
.
442
21.4.2
SUMMARY
.
444
21.5
A
NEW
REPRESENTATION
OF
6-D
CONFORMAL
SPACE
.
445
21.5.1
THE
MULTIPARTICLE
STA
.
447
21.5.2
2-PARTICLE
PAULI
STATES
AND
THE
QUANTUM
CORRELATOR
.
447
21.5.3
A
6-D
REPRESENTATION
IN
THE
MSTA
.
450
21.5.4
LINK
WITH
TWISTORS
.
452
21.5.5
THE
SPECIAL
CONFORMAL
GROUP
.
454
21.5.6
6-D
SPACE
OPERATIONS
.
455
21.6
SUMMARY
AND
CONCLUSIONS
.
456
VII
COMPUTATIONAL
METHODS
IN
CLIFFORD
ALGEBRAS
459
22
CLIFFORD
ALGEBRAS
AS
PROJECTIONS
OF
GROUP
ALGEBRAS
VLADIMIR
M.
CHERNOV
461
22.1
INTRODUCTION
.
461
22.2
GROUP
ALGEBRAS
AND
THEIR
PROJECTION
.
462
22.2.1
BASIC
DEFINITIONS
AND
EXAMPLES
.
462
22.2.2
CLIFFORD
ALGEBRAS
AS
PROJECTIONS
OF
GROUP
ALGEBRAS
.
.
464
22.3
APPLICATIONS
.
465
22.3.1
FAST
ALGORITHMS
OF
THE
DISCRETE
FOURIER
TRANSFORM
.
.
465
22.3.2
FAST
ALGORITHMS
FOR
FIVE
FOURIER
SPECTRA
CALCULATION
IN
THE
ALGEBRA
A(R,
S
3
)
.
466
22.3.3
FAST
ALGORITHM
FOR
THREE
COMPLEX
FOURIER
SPECTRA
WITH
OVERLAPPING
.
470
22.3.4
FAST
ALGORITHMS
FOR
DISCRETE
FOURIER
TRANSFORMS
WITH
MAXIMAL
OVERLAPPING
.
473
22.4
CONCLUSION
.
476
23
COUNTEREXAMPLES
FOR
VALIDATION
AND
DISCOVERING
OF
NEW
THEOREMS
PERTTI
LOUNESTO
477
23.1
INTRODUCTION
.
477
23.2
THE
ROLE
OF
COUNTEREXAMPLES
IN
MATHEMATICS
.
477
23.3
CLIFFORD
ALGEBRAS:
AN
OUTLINE
.
479
23.3.1
THE
CLIFFORD
ALGEBRA
OF
THE
EUCLIDEAN
PLANE
.
479
23.3.2
THE
CLIFFORD
ALGEBRA
OF
THE
MINKOWSKI
SPACE-TIME
.
480
CONTENTS
XV
23.3.3
CLIFFORD
ALGEBRA
VIEWED
BY
MEANS
OF
THE
MATRIX
ALGEBRA
.
481
23.4
PRELIMINARY
COUNTEREXAMPLES
IN
CLIFFORD
ALGEBRAS
.
482
23.5
COUNTEREXAMPLES
ABOUT
SPIN
GROUPS
.
483
23.5.1
COMMENT
ON
BOURBAKI
1959
.
484
23.5.2
EXPONENTIALS
OF
BIVECTORS
.
485
23.5.3
INTERNET
AS
A
SCIENTIFIC
FORUM
.
486
23.5.4
HOW
DID
I
LOCATE
THE
ERRORS
AND
CONSTRUCT
MY
COUNTEREXAMPLES?
.
486
23.5.5
PROGRESS
IN
SCIENCE
VIA
COUNTEREXAMPLES
.
487
23.6
COUNTEREXAMPLES
ON
THE
INTERNET
.
488
24
THE
MAKING
OF
GABLE:
A
GEOMETRIC
ALGEBRA
LEARNING
ENVIRONMENT
IN
MATLAB
STEPHEN
MANN,
LEO
DORST,
AND
TIM
BOUMA
491
24.1
INTRODUCTION
.
491
24.2
REPRESENTATION
OF
GEOMETRIC
ALGEBRA
.
493
24.2.1
THE
MATRIX
REPRESENTATION
OF
GABLE
.
494
24.2.2
THE
REPRESENTATION
MATRICES
.
495
24.2.3
THE
DERIVED
PRODUCTS
.
496
24.2.4
REPRESENTATIONAL
ISSUES
IN
GEOMETRIC
ALGEBRA
.
497
24.2.5
COMPUTATIONAL
EFFICIENCY
.
499
24.2.6
ASYMPTOTIC
COSTS
.
501
24.3
INVERSES
.
503
24.4
MEET
AND
JOIN
.
506
24.4.1
DEFINITION
.
506
24.5
GRAPHICS
.
507
24.6
EXAMPLE:
PAPPUS
'
S
THEOREM
.
508
24.7
CONCLUSIONS
.
509
25
HELMSTETTER
FORMULA
AND
RIGID
MOTIONS
WITH
CLIFFORD
RAFAL
ABLAMOWICZ
512
25.1
INTRODUCTION
.
512
25.2
VERIFICATION
OF
THE
HELMSTETTER
FORMULA
.
513
25.2.1
NUMERIC
EXAMPLE
WHEN
N
=
3
.
516
25.2.2
SYMBOLIC
COMPUTATIONS
WHEN
N
=
3
.
516
25.3
RIGID
MOTIONS
WITH
CLIFFORD
ALGEBRAS
.
518
25.3.1
GROUP
PIN(3)
.
519
25.3.2
GROUP
SPIN(3)
.
524
25.3.3
DEGENERATE
CLIFFORD
ALGEBRA
AND
THE
PROPER
RIGID
MOTIONS
.
529
25.4
SUMMARY
.
532
XVI
CONTENTS
REFERENCES
INDEX
535
583 |
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| illustrated | Illustrated |
| indexdate | 2024-08-17T00:43:41Z |
| institution | BVB |
| isbn | 0817641998 3764341998 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009482120 |
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| physical | XXVI, 592 S. Ill., graph. Darst. |
| publishDate | 2001 |
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| publisher | Birkhäuser |
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| spelling | Geometric Algebra with applications in science and engineering Eduardo Bayro Corrochano ... [Eds.] Boston [u.a.] Birkhäuser 2001 XXVI, 592 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Geometrische algebra gtt Geometry, Algebraic Geometrische Algebra (DE-588)4156707-9 gnd rswk-swf Geometrische Algebra (DE-588)4156707-9 s DE-604 Bayro Corrochano, Eduardo Sonstige (DE-588)123058538 oth DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009482120&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Geometric Algebra with applications in science and engineering Geometrische algebra gtt Geometry, Algebraic Geometrische Algebra (DE-588)4156707-9 gnd |
| subject_GND | (DE-588)4156707-9 |
| title | Geometric Algebra with applications in science and engineering |
| title_auth | Geometric Algebra with applications in science and engineering |
| title_exact_search | Geometric Algebra with applications in science and engineering |
| title_full | Geometric Algebra with applications in science and engineering Eduardo Bayro Corrochano ... [Eds.] |
| title_fullStr | Geometric Algebra with applications in science and engineering Eduardo Bayro Corrochano ... [Eds.] |
| title_full_unstemmed | Geometric Algebra with applications in science and engineering Eduardo Bayro Corrochano ... [Eds.] |
| title_short | Geometric Algebra with applications in science and engineering |
| title_sort | geometric algebra with applications in science and engineering |
| topic | Geometrische algebra gtt Geometry, Algebraic Geometrische Algebra (DE-588)4156707-9 gnd |
| topic_facet | Geometrische algebra Geometry, Algebraic Geometrische Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009482120&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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