Parametric Curves and Applications in CAGD: Introduction and some Special Issues

T. Recio^{∗}  J.R. Sendra & J. Sendra^{†}
Dpto. de Matemáticas  Dpto. de Matemáticas
Universidad de Cantabria  Universidad de Alcalá
Santander 39071 (Spain)  Madrid 28871 (Spain)
recio@matsun1.unican.es  mtsendra@alcala.es

Abstract

In this talk we briefly introduce the notion and main problems related to parametric curves, and we focus on two issues specially interesting in applications: the problem of computing real reparametrization of real parametric curves, and the rationality analysis and parametrization of offset curves of parametric curves.

Parametric curves play an important role in the field of Computer Aided Geometric Design. They arise in practical applications as computer graphics, geometric modeling or manipulation of offset varieties. The natural field of applications of rational curves is Computer Aided Design. Indeed, real varietes are the most common ones in Geometric Design. However, since Algebraic Geometry is mainly developed over algebraically closed fields, the majority of the previous algorithms are checked only over the complex numbers. We present an algorithm that decides whether a complex rational parametrization of a plane curve can be reparametrized over the reals, and if so, it computes a real parametrization of the curve.

An interesting problem to mention is the algebraic manipulation of offset curves. Offset curves of a given curve, or more formerly parallel curves, are defined as the envelopes of systems of circles with centres on the original curve and fixed radious. They arise in practical applications as tolerance analysis, geometric control, robot path-planning and numerical-control machining problems. One of the main problems in offset curves manipulation is that the rationality of the original curve is not preserved (in general) when the offset curve is considered. In this paper, a rationality analysis of the offset curves is presented, and symbolic algorithms to parametrized the rational components of the offset curves to parametric curves are derived.

keywords: parametric curves, cagd, introduction, real parametrization, real curves, offset curves, rational offset curves.

∗Partially supported by CICyT PB 92/0498/C02/01 (Geometría Real y Algoritmos), Esprit/Bra 6846 (Posso), and TIC-1026-CE

†Partially supported by Univ. Alcalá Proy. 030/95