Minkowski isoperimetric-hodograph curves
| Autor: | M. A. Slawinski, Rachid Ait-Haddou, Luc Biard |
|---|---|
| Přispěvatelé: | University of Calgary, Laboratoire de Modélisation et Calcul (LMC - IMAG), Université Joseph Fourier - Grenoble 1 (UJF)-Institut National Polytechnique de Grenoble (INPG)-Centre National de la Recherche Scientifique (CNRS) |
| Rok vydání: | 2000 |
| Předmět: |
Dual Bézier curves
Frenet–Serret formulas Minkowski's theorem Aerospace Engineering 02 engineering and technology 01 natural sciences Minkowski space 0202 electrical engineering electronic engineering information engineering Involute-evolute Mathematics::Metric Geometry Pythagorean-hodograph curves 0101 mathematics Minkowski geometry Mathematics Classification of electromagnetic fields 010102 general mathematics Mathematical analysis NC milling Serret–Frenet equations 020207 software engineering 16. Peace & justice Computer Graphics and Computer-Aided Design Frenet frame Minkowski plane Differential geometry [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] Modeling and Simulation Automotive Engineering Family of curves Rational offset curves Isoperimetric inequality |
| Zdroj: | Computer Aided Geometric Design Computer Aided Geometric Design, Elsevier, 2000, 17 (9), pp.835-861. ⟨10.1016/S0167-8396(00)00029-7⟩ |
| ISSN: | 0167-8396 |
| DOI: | 10.1016/s0167-8396(00)00029-7 |
| Popis: | International audience; General offset curves are treated in the context of Minkowski geometry, the geometry of the two-dimensional plane, stemming from the consideration of a strictly convex, centrally symmetric given curve as its unit circle. Minkowski geometry permits us to move beyond classical confines and provides us with a framework in which to generalize the notion of Pythagorean-hodograph curves in the case of rational general offsets, namely, Minkowski isoperimetric-hodograph curves. Differential geometric topics in the Minkowski plane, including the notion of normality, Frenet frame, Serret–Frenet equations, involutes and evolutes are introduced. These lead to an elegant process from which an explicit parametric representation of the general offset curves is derived. Using the duality between indicatrix and isoperimetrix and between involutes and evolutes, rational curves with rational general offsets are characterized. The dual Bézier notion is invoked to characterize the control structure of Minkowski isoperimetric-hodograph curves. This characterization empowers the constructive process of freeform curve design involving offsetting techniques. |
| Databáze: | OpenAIRE |
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