On computing structural changes in evolving surfaces and their appearance

摘要

As a surface undergoes a one-parameter family of deformations, its shape and its appearance change smoothly except at certain critical parameter values where abrupt structural changes occur. This paper considers the case of surfaces defined as the zero set of smooth density functions undergoing a Gaussian diffusion process and addresses the problem of computing the critical parameter values corresponding to structural changes in the parabolic curves of a surface and in its aspect graph. An algorithm based on homotopy continuation 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.