Computing the Distance between Piecewise-Linear Bivariate Functions

Abstract

We consider the problem of computing the distance between two piecewise-linear bivariate functions f and g defined over a common domain M , induced by the L 2 norm—that is, ‖ f - g ‖2 = √∫ M ( f - g ) 2 . If f is defined by linear interpolation over a triangulation of M with n triangles and g is defined over another such triangulation, the obvious naive algorithm requires Θ( n 2 ) arithmetic operations to compute this distance. We show that it is possible to compute it in O ( n log 4 n log log n ) arithmetic operations by reducing the problem to multipoint evaluation of a certain type of polynomials. We also present several generalizations and an application to terrain matching.