Pythagorean-Hodograph Curves: Algebra and Geometry InseparableSpringer Science & Business Media, 2008 M02 1 - 728 pages 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. |
Contents
| 1 | |
| 4 | |
Polynomials | 29 |
Complex Numbers 45 | 44 |
Quaternions | 61 |
Clifford Algebra | 79 |
Coordinate Systems 89 | 88 |
Differential Geometry | 131 |
Complex Representation | 407 |
Rational Pythagoreanhodograph Curves | 427 |
Pythagorean Hodographs in R3 455 | 454 |
Quaternion Representation | 469 |
Helical Polynomial Curves | 485 |
Minkowski Pythagorean Hodographs | 507 |
Planar Hermite Interpolants | 523 |
Elastic Bending Energy 543 | 542 |
Algebraic Geometry | 197 |
NonEuclidean Geometry | 231 |
11 | 248 |
Numerical Stability | 261 |
16 | 369 |
Tschirnhausens Cubic 393 | 392 |
CCR0202179 DMS0138411 CCR9902669 DMI9908525 CCR9530741 | 394 |
Other editions - View all
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable Rida T Farouki No preview available - 2009 |
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable Rida T Farouki No preview available - 2010 |
Common terms and phrases
algebraic curve algorithm angle arc length arithmetic B–spline barycentric coordinates basis functions Bernstein form Bézier curve Bézier form Cartesian coordinates circle coefficients complex numbers components compute condition number consider control points control polygon corresponding curvilinear coordinates cusp defined degree degree–n derivatives differential distance double point equation error evolute example expression floating–point geodesic geometrical given curve hodograph homogeneous coordinates identify integral interpolation intersection interval involutes knots linear loci locus matrix multiplicity non–zero normal obtain one–to–one orthogonal osculating circle parabola parameter parameterization perturbations PH curves planar plane curve polynomial curve polynomial p(t problem projective properties Pythagorean quadratic quartic quaternions rational curves rational functions real numbers real roots representation rotation satisfy scalar segments self–intersections singular points solution space curve specified spline tangent line tensor theorem three–dimensional transformation unit vector untrimmed offset values vanishes variable