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

提出的研究是将共形面理论应用于各种几何表示来计算共形结构。使用计算保形结构，可以开发新的几何表示和分析工具，这将为科学和工程的多个前沿的进步铺平道路。本文提出的工作将特别探索这些在计算机图形学和可视化方面的潜力。构建上保形结构将许多三维（3D）几何问题转换为二维（2D），并为许多基本几何问题提供了有效的方法。这些方法可以立即受益于广泛的应用，如表面分类，表面匹配和形状分析，几何建模，仿真，图形渲染和可视化。执行保形参数化需要解决大型最小二乘问题。为了进一步推动保形结构在交互或时间关键操作中的应用，研究小组将研究基于稀疏QR和不完全稀疏QR算法的最小二乘问题的新颖迭代方法。各种科学和工程应用涉及关键操作，如建模、设计、分析、仿真和图形渲染。所有这些操作都建立在几何表示的基础之上。在这个项目中，科学家们希望通过引入一种独特的几何表面特征的共形结构来彻底改变这种基础。许多物理定律都受保形结构的支配。例如，表面上的热扩散和电磁场分布，肥皂泡中的张力以及理论物理中弦理论的部分内容都是由共形表面结构决定的。在现有成功的鼓舞下，科学家们努力探索和揭示保形结构在计算机图形学、几何建模和许多科学计算中的潜力。为了说明这种潜力，考虑保形结构的一个方面，即一个表面的标准扁平化成为一个平面，导致看似复杂的三维几何图形的图像表示。总的来说，在这个项目中开发的工具可以帮助促进有效技术的发展，以处理与科学模拟、数据探索、身份匹配或形状分析相关的新问题，用于监视和生物发现。

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.