Computing intersection and self-intersection loci of parametrized surfaces using regular systems and Gröbner bases

This paper presents two general and efficient methods for determining intersection and self-intersection loci of rationally parametrized surfaces. One of the methods, based on regular systems, is capable of computing the exact parametric loci of intersection and self-intersection. The other, based on Gröbner bases, can compute the minimal varieties passing through the exact parametric loci. The relation between the results computed by the two methods is established and algorithms for computing parametric loci of intersection and self-intersection are described. Experimental results and comparisons with some existing methods show that our algorithms have a good performance.

► We present two methods for computing the parametric loci of (self-)intersection. ► The regular system method can determine exact parametric loci. ► The Gröbner basis method can determine minimal varieties. ► We prove that the minimal variety is the Zariski closure of the exact parametric locus. ► Experimental results and comparisons show that our methods have a good performance.