A simple method for implicitizing rational curves and surfaces

This paper presents 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 ui, 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 ui with constant coefficients, and finally solving the linear system for ui. 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.