Shape-Preserving Meshes

Smooth surfaces are approximated by discrete triangle meshes for applications in computer graphics. Various discrete operators have been proposed for estimating differential quantities of triangle meshes, such as curvatures, for geometric processing tasks. Since a smooth surface can be approximated by many different triangle meshes, we propose to investigate which triangle mesh yields an estimation of differential quantities with optimal accuracy and how to compute such an optimal triangle mesh approximating the given smooth surface. Specifically, we study a special type of triangle meshes, called shapepreserving meshes, that preserve the local shapes of the smooth surface they represent, and characterize optimal shape-preserving meshes. We present an efficient method for computing the so called optimal shape-preserving meshes, and prove the convergence of several discrete differential operators on optimal shape-preserving meshes, an important property that does not hold for general triangle meshes. We also show that shape-preserving meshes lead to more accurate estimation of surface differential quantities as compared with other general triangle meshes obtained by commonly used surface meshing methods for the same smooth surface.