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. It emphasizes the interplay of ideas from algebra and geometry and their historical origins and includes many figures, worked examples, and detailed algorithm descriptions.

Hersteller:
Springer Verlag GmbH
juergen.hartmann@springer.com
Tiergartenstr. 17
DE 69121 Heidelberg

Preface Chapter 1: Introduction PART I: ALGEBRA Chapter 2: Preamble Chapter 3: Polynomials Chapter 4: Complex numbers Chapter 5: Quaternions Chapter 6: Clifford algebra PART II: GEOMETRY Chapter 7: Coordinate systems Chapter 8: Differential geometry Chapter 9: Algebraic geometry Chapter 10: Non-Euclidean geometry PART III: COMPUTER-AIDED GEOMETRIC DESIGN Chapter 11: The Bernstein basis Chapter 12: Numerical stability Chapter 13: Bezier curves and surfaces Chapter 14: C^2 cubic spline curves Chapter 15: Spline basis functions PART IV: PLANAR PYTHAGOREAN-HODOGRAPH CURVES Chapter 16: Arc-length parameterization Chapter 17: Pythagorean-hodograph curves Chapter 18: Tschirnhausen''s cubic Chapter 19: Complex representation Chapter 20: Rational Pythagorean-hodograph curves PART V: SPATIAL PYTHAGOREAN-HODOGRAPH CURVES Chapter 21: Pythagorean hodographs in R^3 Chapter 22: Quaternion representation Chapter 23: Helical polynomial curves Chapter 24: Minkowski Pythagorean hodographs PART VI: ALGORITHMS Chapter 25: Planar Hermite interpolants Chapter 26: Elastic bending energy Chapter 27: Planar C^2 PH quintic splines Chapter 28: Spatial Hermite interpolants PART VII: APPLICATIONS Chapter 29: Real-time CNC interpolators Chapter 30: Rotation-minimizing frames Chapter 31: Closure References Index