Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable

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Springer Science & Business Media, 2007 M10 11 - 728 pages
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By virtue of their special algebraic structures, Pythagorean-hodograph (PH) curves offer unique advantages for computer-aided design and manufacturing, robotics, motion control, path planning, computer graphics, animation, and related fields. This book offers a comprehensive and self-contained treatment of the mathematical theory of PH curves, including algorithms for their construction and examples of their practical applications. Special features include an emphasis on the interplay of ideas from algebra and geometry and their historical origins, detailed algorithm descriptions, and many figures and worked examples. The book may appeal, in whole or in part, to mathematicians, computer scientists, and engineers.

 

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Contents

comprising a number of chapters Parts I through III are expository in nature
1
Preamble
11
Polynomials
29
Complex Numbers 45
44
Quaternions
61
Clifford Algebra
79
Coordinate Systems 89
88
Differential Geometry
131
Tschirnhausens Cubic 393
392
CCR0202179 DMS0138411 CCR9902669 DMI9908525 CCR9530741
394
Complex Representation
407
Rational Pythagoreanhodograph Curves
427
Pythagorean Hodographs in R3 455
454
Quaternion Representation
469
Helical Polynomial Curves
485
Minkowski Pythagorean Hodographs
507

Algebraic Geometry
197
NonEuclidean Geometry
231
The Bernstein Basis 249
248
Numerical Stability
261
Bezier Curves and Surfaces
295
C2 Cubic Spline Curves
323
Spline Basis Functions 345
344
Arclength Parameterization
369
Pythagoreanhodograph Curves
381
Planar Hermite Interpolants
523
Elastic Bending Energy 543
542
Planar C2 PH Quintic Splines
555
Spatial Hermite Interpolants
595
Realtime CNC Interpolators
619
Rotationminimizing Frames
661
Closure 693
692
Index
719
Copyright

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