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

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The proposed research is to apply conformal surface theory tovarious geometry representations to compute conformalstructures. Using computed conformal structures, new geometryrepresentation and analysis tools can be developed, which will pavethe road for advances in multiple fronts of science andengineering. The work proposed herein will especially explore thesepotentials in computer graphics and visualization. Building upconformal structures recasts many three dimensional (3D) geometricproblems into two dimensions (2D) and leads to efficient approachesfor a number of fundamental geometric problems. These approaches canthen immediately benefit a wide range of applications, such assurface classification, surface matching and shape analysis,geometric modeling, simulation, graphics rendering andvisualization. Performing conformal parameterization requiressolving large least squares problems. To further push theapplication of the conformal structure to interactive or timecritical operations, the research team will investigate noveliterative methods for least-squares problems based on sparse QR andincomplete sparse QR algorithms.Various scientific and engineering applications concern about keyoperations such as modeling, design, analysis, simulation, andgraphics rendering. All these operations are built on top afoundation of geometry representation. In this project, scientistsaim to revolutionize this foundation by introducing a conformalstructure uniquely characterizing geometry surfaces. Many laws ofphysics are governed by conformal structures. For example, heatdiffusion and electromagnetic field distribution on surfaces,tension in soap bubbles and parts of string theory in theoreticalphysics are determined by conformal surface structures. Encouragedby the existing success, the scientists strive to explore and unveilthe potentials of conformal structures for computer graphics,geometric modeling and much of scientific computing. To illustratethis potential, consider one aspect of conformal structures, namelythe canonical flattening of a surface into a plane, resulting in animage like representation of seemingly complicated three dimensionalgeometry. Overall, the tools developed in this project can helpboost the development of effective techniques to deal with theemerging problems related to scientific simulation, dataexploration, and identity matching or shape analysis forsurveillance and biological discovery.