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

提出的研究是将共形曲面理论应用于各种 几何 表示 到 计算 保形构造 使用计算机 保形结构，新的几何表示和分析工具可以开发，这将铺平道路， 前进之路 在多个 战线 科学和工程。本文提出的工作将特别探索这些潜力在计算机图形学和可视化。 共形结构的建立将许多三维几何问题转化为二维几何问题，并为许多基本几何问题提供了有效的解决途径。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、几何的 建模， 仿真， 图形 渲染 和可视化。 执行 保角参数化 需要解决大型 最小二乘 问题 进一步 为了将共形结构应用于交互式或时间临界操作，研究团队将研究基于稀疏QR和不完全稀疏QR算法的最小二乘问题的新迭代方法。各种科学和工程应用涉及建模，设计，分析，仿真和图形绘制等关键操作。 所有这些操作都是建立在几何表示的基础之上的.在这个项目中，科学家们通过引入一种独特的几何表面特征的保形结构来彻底改变这一基础。许多物理定律都是由共形结构决定的。 例如，热扩散和电磁 表面上的场分布、肥皂泡中的张力以及理论物理学中的部分弦理论都是由共形表面结构决定的。受到现有成功的鼓舞，科学家们努力探索和揭示计算机图形学，几何建模和许多科学计算的共形结构的潜力。为了说明这种可能性，考虑共形结构的一个方面，即表面到平面的正则平坦化，导致看似复杂的三维几何的图像表示。 总体而言，本项目开发的工具有助于促进有效技术的发展，以应对新兴的 问题 相关 到 科学 仿真， 数据探索， 和 标识匹配 或 形状分析 for surveillance监视and biological生物discovery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.