Computing Discrete Shape Operators on General Meshes

Eitan Grinspun, Yotam Gingold, Jason Reisman, Denis Zorin
Computer Graphics Forum 25(3):547–556. Also in Proceedings of Eurographics 2006.
Best Paper Award, 3rd

Paper: PDF (2.1M)


Discrete curvature and shape operators, which capture complete information about directional curvatures at a point, are essential in a variety of applications: simulation of deformable two-dimensional objects, variational modeling and geometric data processing. In many of these applications, objects are represented by meshes. Currently, a spectrum of approaches for formulating curvature operators for meshes exists, ranging from highly accurate but computationally expensive methods used in engineering applications to efficient but less accurate techniques popular in simulation for computer graphics.

We propose a simple and efficient formulation for the shape operator for variational problems on general meshes, using degrees of freedom associated with normals. On the one hand, it is similar in its simplicity to some of the discrete curvature operators commonly used in graphics; on the other hand, it passes a number of important convergence tests and produces consistent results for different types of meshes and mesh refinement.


 author    = {Grinspun, Eitan and Gingold, Yotam and Reisman, Jason and Zorin, Denis},
 title     = {Computing discrete shape operators on general meshes},
 journal   = {Computer Graphics Forum (Eurographics)},
 year      = {2006},
 volume    = {25},
 number    = {3},
 pages     = {547--556},
 issn      = {1467-8659},
 doi       = {10.1111/j.1467-8659.2006.00974.x}

An addendum:

A Note on the Triangle-Centered Quadratic Interpolation Discretization of the Shape Operator

Jason Reisman, Eitan Grinspun, Denis Zorin
Technical Report, New York University, 2007.

Paper (PDF, 0.8M)


In this note we consider a simple shape operator discretization for general meshes, based on computing an interpolating quadratic function passing through vertices of a triangle and its edge-adjacent neighbors. This approximation is computationally simple and consistent for a broad class of meshes. However, its convergence properties in the context of mesh optimization problems are not as good as some of the previously proposed techniques and it suffers from instabilities for certain point configurations.


  author      = {Jason Reisman, Eitan Grinspun, Denis Zorin},
  title       = {A note on the triangle-centered quadratic interpolation discretization of the shape operator},
  institution = {New York University},
  year        = {2007},