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>F</f> of estimated degree with indeterminate coefficients <f>ui</f>, substituting the rational expressions for the given parametric curve or surface into <f>F</f> to yield a rational expression <f>g/h</f> in the parameter <f>s</f> (or <f>s</f> and <f>t</f>), equating the coefficients of <f>g</f> in terms of <f>s</f> (and <f>t</f>) to 0 to generate a sparse, partially triangular system of linear equations in <f>ui</f> with constant coefficients, and finally solving the linear system for <f>ui</f>. If a nontrivial solution is found, then an implicit polynomial is obtained; otherwise, one repeats the same process, increasing the degree of <f>F</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. [Copyright &y& Elsevier]