Yaron Lipman (Computer Science, Tel Aviv Univ.)
CGGC Seminar: 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.