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Journal Articles Journal of Symbolic Computation Year : 2004

A Simple Method for Implicitizing Rational Curves and Surfaces

Abstract

This paper presents a simple method for converting rational parametric equations of curves and surfaces into implicit equations. The method proceeds by writing out the implicit polynomial $F$ of estimated degree with indeterminate coefficients $u_i$, substituting the rational expressions of the given parametric curve or surface into $F$ to yield a rational expression $g/h$ in the parameter $s$ (or $s$ and $t$), equating the coefficients of $g$ with respect to $s$ (and $t$) to 0 to generate a sparse, partially triangular system of linear equations in $u_i$ with constant coefficients, and finally solving the linear system for $u_i$. If a nontrivial solution is found, then an implicit polynomial is obtained; otherwise, one repeats the same process by increasing the degree of $F$. Our experiments show that this simple method is efficient. It performs particularly well in the presence of base points and may detect the dependency of parameters incidentally.

Dates and versions

inria-00099943 , version 1 (26-09-2006)

Identifiers

Cite

Dongming Wang. A Simple Method for Implicitizing Rational Curves and Surfaces. Journal of Symbolic Computation, 2004, 38 (1), pp.899-914. ⟨10.1016/j.jsc.2004.02.004⟩. ⟨inria-00099943⟩
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