Pythagorean-hodograph curves. Algebra and geometry inseparable
A polynomial curve \(c:{\mathbb R}\to{\mathbb R}^k\) is a Pythagorean-Hodograph (PH) curve, if and only if the scalar product \(\langle {dc\over dt},{dc\over dt}\rangle\) is a square in the polynomial ring \({\mathbb R}[t]\); the cases \(k=2,3\) correspond to Pythagorean triples and quadruples in \({\mathbb R}[t]\). Such curves have a polynomial arc length function and a rational unit vector field, which is responsible for applications in computer-aided geometric design. This comprehensive and self-contained volume gives a detailed and complete over\-view of our current knowledge of PH curves, their generalizations and applications. The first three introductory chapters extensively deal with algebraic basics (including quaternions and Clifford algebra), geometry, and computer-aided geometric design (Bézier/spline curves and surfaces, numerical stability). Chapters IV and V introduce planar and spatial PH curves and their generalizations (arc-length parametrizations, Tschirnhausen's cubic, complex representation, rational PH curves, quaternion representations, helical polynomial curves, Minkowski Pythagorean hodographs). Chapter VI on algorithms focuses on classes of PH curves capable of Hermite interpolation. Further applications -- real-time CNC interpolators and rotation-minimizing frames -- are given in the last chapter.
scientific article; zbMATH DE number 5189553