Using multivariate resultants to find the intersection of three quadric surfaces

Abstract

Macaulay's concise but explicit expression for multivariate resultants has many potential applications in computer-aided geometric design. Here we describe its use in solid modeling for finding the intersections of three implicit quadric surfaces. By Bezout's theorem, three quadric surfaces have either at most eight or infinitely many intersections. Our method finds the intersections, when there are finitely many, by generating a polynomial of degree at most eight whose roots are the intersection coordinates along an appropriate axis. Only addition, subtraction, and multiplication are required to find the polynomial. But when there are possibilities of extraneous roots, division and greatest common divisor computations are necessary to identify and remove them.