Intriguing Relation between Geometry and Mesh Topology and Its Effects on Subdivision Modeling

This webpage provides information about intriguing relation between geometry and mesh topology and its effects on subdivision modeling. Click here to access an html file that provide easy access to several flash videos. These screen-captured voice-over videos show how to create these models using TopMod. Click here to see an informal powerpoint presentation by Ergun Akleman in November 2005 given in a Brownbag series to get a feedback from John Keyser, Don House, Fred Parke and others. It provides a background story about how we get the result. Click here to download TopMod. After the submission of the paper, we have discovered Gauss-Bonnet theorem. After reading about it in detail we have realized that it can be a good idea to include a discussion about the relationship with Gauss-Bonnet theorem. We are planning to include a discussion like the following to the final version of the paper. (See paper) Our theorem is very similar to the Gauss-Bonnet Theorem that says that the integral of the Gaussian curvature over a closed smooth surface is equal to 2\pi times the Euler characteristic of the surface which is 2-2g. In fact, for triangular meshes Calladine introduced a discrete version of Gaussian curvature that uses angular deviation. Calladine's discrete Gaussian curvature given as (Angular Deviation)/(Area Associated with Vertex) Despite the similarity, angle deviations are different than Gaussian curvature and its discrete versions. For instance, unlike sum of angle deviations, sum of discrete Gaussian Curvature will not be equal to 2pi (2-2g). Both Gaussian curvature and angle deviations are rotation and translation invarient. Angle deviation is also scale invarient which can make it useful for shape retrieval.