Discrete Differential Geometry and its Applications

Yiying Tong

Postdoctoral Scholar
California Institute of Technology

January 8, 2006
Talk: 10:30 am - 11:30 am
3105 Engineering

Host: Anil K. Jain

In this talk, we demonstrate the value of a structure-preserving discretization of geometry. We first present a general framework for calculus on manifolds represented as meshes. The framework is built on a formal discretization of Cartan's exterior calculus of differential forms. Then we point out its relationship to commonly-used geometric computational tools like discrete Laplacian operators, Hodge decomposition, and mean curvature flow to name a few. Applying this general framework to geometric modeling and texture mapping, we show various algorithms for geometric texture synthesis, quadrangulation of triangular meshes, and seamless texturing of arbitrary surfaces from photos. With the exact same framework, we demonstrate how fluid simulation on simplicial complexes can be implemented in an intrinsic manner through proper discretization of flux and vorticity. Extending the framework to general dynamics, we finally show how the preservation of geometric structures directly leads to numerically-superior time integrators.

Yiying Tong is a postdoctoral scholar in computer science department at California Institute of Technology. He received a Master's degree from Zhejiang University in China, and a PhD degree in computer science from University of Southern California. His research interests include: discrete differential geometry, computer animation, and discrete geometric modeling. He currently focuses on discrete differential forms and their application in meshing, fluid simulation, and elasticity.