MSPA-MCS: Collaborative Research: Computer Graphics and Visualization Using Conformal Geometry

The proposed research is to apply conformal surface theory to

various geometry representations to compute conformal

structures. Using computed conformal structures, new geometry

representation and analysis tools can be developed, which will pave

the road for advances in multiple fronts of science and

engineering. The work proposed herein will especially explore these

potentials in computer graphics and visualization. Building up

conformal structures recasts many three dimensional (3D) geometric

problems into two dimensions (2D) and leads to efficient approaches

for a number of fundamental geometric problems. These approaches can

then immediately benefit a wide range of applications, such as

surface classification, surface matching and shape analysis,

geometric modeling, simulation, graphics rendering and

visualization. Performing conformal parameterization requires

solving large least squares problems. To further push the

application of the conformal structure to interactive or time

critical operations, the research team will investigate novel

iterative methods for least-squares problems based on sparse QR and

incomplete sparse QR algorithms.

Various scientific and engineering applications concern about key

operations such as modeling, design, analysis, simulation, and

graphics rendering. All these operations are built on top a

foundation of geometry representation. In this project, scientists

aim to revolutionize this foundation by introducing a conformal

structure uniquely characterizing geometry surfaces. Many laws of

physics are governed by conformal structures. For example, heat

diffusion and electromagnetic field distribution on surfaces,

tension in soap bubbles and parts of string theory in theoretical

physics are determined by conformal surface structures. Encouraged

by the existing success, the scientists strive to explore and unveil

the potentials of conformal structures for computer graphics,

geometric modeling and much of scientific computing. To illustrate

this potential, consider one aspect of conformal structures, namely

the canonical flattening of a surface into a plane, resulting in an

image like representation of seemingly complicated three dimensional

geometry. Overall, the tools developed in this project can help

boost the development of effective techniques to deal with the

emerging problems related to scientific simulation, data

exploration, and identity matching or shape analysis for

surveillance and biological discovery.