A general framework for surface modeling using geometric partial differential equations

Abstract

In this paper, a general framework for surface modeling using geometric partial differential equations (PDEs) is presented. Starting with a general integral functional, we derive an Euler–Lagrange equation and then a geometric evolution equation (also known as geometric flow). This evolution equation is universal, containing several well-known geometric partial differential equations as its special cases, and is discretized under a uniform framework over surface meshes. The discretization of the equation involves approximations of curvatures and several geometric differential operators which are consistently discretized based on a quadratic fitting scheme. The proposed algorithm can be used to construct surfaces for geometric design as well as simulate the behaviors of various geometric PDEs. Comparative experiments show that the proposed approach can handle a large number of geometric PDEs and the numerical algorithm is efficient.