Geometric Methods in Engineering Applications
Book Chapter in "Mathematics and Computation, a Contemporary View" 2008
Xianfeng Gu Yalin Wang Hsiao-Bing Cheng Li-Tien Cheng and Shing-Tung Yau
In this work, we introduce two set of algorithms
inspired by the ideas from modern geometry. One is computational
conformal geometry method, including harmonic maps,
holomorphic 1-forms and Ricci flow. The other one is optimization
method using affine normals.
In the first part, we focus on conformal geometry. Conformal
structure is a natural structure of metric surfaces. The concepts
and methods from conformal geometry play important roles
for real applications in scientific computing, computer graphics,
computer vision and medical imaging fields.
This work systematically introduces the concepts, methods
for numerically computing conformal structures inspired by
conformal geometry. The algorithms are theoretically rigorous
and practically efficient.
We demonstrate the algorithms by real applications, such as
surface matching, global conformal parameterization, conformal
brain mapping etc.
In the second part, we consider minimization of a real-valued
function f over Rn+1 and study the choice of the affine normal of
the level set hypersurfaces of f as a direction for minimization.
The affine normal vector arises in affine differential geometry
when answering the question of what hypersurfaces are invariant
under unimodular affine transformations. It can be computed at
points of a hypersurface from local geometry or, in an alternate
description, centers of gravity of slices. In the case where f is
quadratic, the line passing through any chosen point parallel to its
affine normal will pass through the critical point of f . We study
numerical techniques for calculating affine normal directions of
level set surfaces of convex f for minimization algorithms.