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A Simple Method for Implicitizing Rational Curves and Surfaces

Dongming Wang 1, 2
2 SPACES - Solving problems through algebraic computation and efficient software
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
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.
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Submitted on : Tuesday, September 26, 2006 - 10:12:46 AM
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Dongming Wang. A Simple Method for Implicitizing Rational Curves and Surfaces. Journal of Symbolic Computation, Elsevier, 2004, 38 (1), pp.899-914. ⟨10.1016/j.jsc.2004.02.004⟩. ⟨inria-00099943⟩



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