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A simple method for implicitizing rational curves and surfaces. (English) Zbl 1137.65322

Summary: A simple method for converting rational parametric equations of curves and surfaces into implicit equations. The method proceeds via writing out the implicit polynomial \(F\) of estimated degree with indeterminate coefficients \(u_{i}\), substituting the rational expressions for 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 \(g\) in terms of \(s\) (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, 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.

MSC:

65D17 Computer-aided design (modeling of curves and surfaces)
14Q10 Computational aspects of algebraic surfaces
68W30 Symbolic computation and algebraic computation

Software:

Epsilon
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Full Text: DOI

References:

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