Ebook: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Author: Rida T. Farouki (auth.)
- Tags: Computational Mathematics and Numerical Analysis, Geometry, Algebra, Computer Imaging Vision Pattern Recognition and Graphics, Computational Intelligence, Automotive Engineering
- Series: Geometry and Computing 1
- Year: 2008
- Publisher: Springer-Verlag Berlin Heidelberg
- Edition: 1
- Language: English
- pdf
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.
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 will appeal, in whole or in part, to mathematicians, computer scientists, and engineers.
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 will appeal, in whole or in part, to mathematicians, computer scientists, and engineers.
Content:
Front Matter....Pages I-XVI
Introduction....Pages 1-8
Preamble....Pages 11-28
Polynomials....Pages 29-43
Complex Numbers....Pages 45-60
Quaternions....Pages 61-77
Clifford Algebra....Pages 79-86
Coordinate Systems....Pages 89-129
Differential Geometry....Pages 131-196
Algebraic Geometry....Pages 197-230
Non—Euclidean Geometry....Pages 231-246
The Bernstein Basis....Pages 249-260
Numerical Stability....Pages 261-293
B?zier Curves and Surfaces....Pages 295-322
Spline Basis Functions....Pages 323-343
Arc—length Parameterization....Pages 345-366
Pythagorean—hodograph Curves....Pages 369-380
Tschirnhausen's Cubic....Pages 381-391
Complex Representation....Pages 393-406
Rational Pythagorean-hodograph Curves....Pages 407-425
Quaternion Representation....Pages 427-452
Helical Polynomial Curves....Pages 455-467
Minkowski Pythagorean Hodographs....Pages 469-484
Planar Hermite Interpolants....Pages 485-506
Elastic Bending Energy....Pages 507-519
Spatial Hermite Interpolants....Pages 523-541
Real—time CNC Interpolators....Pages 543-553
Rotation—minimizing Frames....Pages 555-594
Closure....Pages 595-615
Back Matter....Pages 619-660
....Pages 661-691