Pythagorean-Hodograph Curves: Algebra and Geometry InseparableSpringer Science & Business Media, 2007 M10 11 - 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
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 |
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 algorithm angle applied approximation arc length arithmetic basis Bernstein called Cartesian circle coefficients complex components compute condition consider constant construction continuity control points coordinates corresponding cubic curvature defined derivatives described determined different direction distance distinct double elements equation error example expression fact factor first follows four functions geometrical given gives hence hodograph homogeneous identify indicate integral interpolation interpretation intersection interval knots known linear matrix means method multiplicity namely nature normal Note obtain offset origin orthogonal parallel parameter parameterization PH curves PH quintic planar plane polygon polynomial positive problem projective quadratic quaternion rational regarded representation respect result roots rotation satisfy segments shape side simple singular smooth solution space curve spatial specified specify spline surface tangent transformation unique unit values variable vector yields
Popular passages
Page vi - Read not to contradict and confute, nor to believe and take for granted, nor to find talk and discourse, but to weigh and consider. Some books are to be tasted, others to be swallowed, and some few to be chewed and digested; that is, some books are to be read only in parts; others to be read but not curiously (carefully), and some few to be read wholly, and with diligence and attention.