On computing structural changes in evolving surfaces and their appearance

As a surface undergoes a one-parameter family of deformations, its shape an d its appearance change smoothly except at certain critical parameter value s where abrupt structural changes occur. This paper considers the case of s urfaces defined as the zero set of smooth density functions undergoing a Ga ussian diffusion process and addresses the problem of computing the critica l parameter values corresponding to structural changes in the parabolic cur ves of a surface and in its aspect graph. An algorithm based on homotopy co ntinuation and curve tracing is proposed in the case of polynomial density functions, whose zero set is an algebraic surface. It has been implemented and examples are presented.