APPLIED MATH SEMINAR

Title: "Geometric Deformations"

Speaker: Yaron Lipman, Tel Aviv University

When/where: Tuesday, February 26th, 4:15 P.M., AKW 200

Deforming objects in a "shape preserving" manner is a challenging problem with various applications in geometric modeling and computer graphics. In this talk I will examine two geometric-driven approaches to the problem.

First, reformulating the problem using the notion of Cartan's moving frames reveals an intriguing relationship between harmonic maps into the group of rotations and shape preserving deformations. The moving frames are known in differential geometry for their ability to simplify some surface theory argumentation. Their employment leads to rather simple algorithms for shape deformation and shape blending of discrete surfaces (meshes). I will present the moving frames as rigid-motion invariant surface representation which is suited for deformations and shape blending. Then I will discuss the optimal rotation field between two isometric surfaces and its use in the context of surface deformation. Second, I will present a recent result demonstrating a closed-form solution to shape preserving space deformations. The new scheme guarantees a pure conformal mapping in 2D and quasi-conformal mappings in 3D. This generalizes recent interesting affine-invariant free form deformation techniques and provides an extremely fast algorithm for shape preserving deformations.