The surface/surface intersection problem by means of matrix based representations

Evaluating the intersection of two rational parameterized algebraic surfaces is an important problem in solid modeling. In this paper, we make use of some generalized matrix based representations of parameterized surfaces in order to represent the intersection curve of two such surfaces as the zero set of a matrix determinant. As a consequence, we extend to a dramatically larger class of rational parameterized surfaces, the applicability of a general approach to the surface/surface intersection problem due to J. Canny and D. Manocha. In this way, we obtain compact and efficient representations of intersection curves allowing to reduce some geometric operations on such curves to matrix operations using results from linear algebra.

► We address the problem of determining the intersection curve of two rational parameterized surfaces. ► General matrix representations of rational parameterized surfaces are used. ► We characterize the intersection curve in terms of the spectrum of a bivariate pencil of matrices. ► A reduction algorithm to extract the continuous part of a bivariate pencil of matrices is given.