Pythagorean-hodograph curves. Algebra and geometry inseparable
type: Item
instance of: scholarly article
title: Pythagorean-hodograph curves. Algebra and geometry inseparable
zbMATH Open document ID: 1144.51004
author: Rida T. Farouki
published in: Geometry and Computing
publication date: 2007-09-11T00:00:00Z
review text: 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.
reviewed by: J. Wallner
Mathematics Subject Classification ID: 51N20
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 15A66
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 65-01
Mathematics Subject Classification ID: 51-01
Mathematics Subject Classification ID: 68U07
zbMATH DE Number: 5189553
zbMATH Keywords: Pythagorean hodograph
zbMATH Keywords: computer aided gometric design
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5189553