Computational and Conformal Geometry

计算和共形几何

• 批准号: 0705455
• 负责人: Christopher Bishop
• 金额: $ 18万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0705455&HistoricalAwards=false
• 关键词: Computational; Conformal; Geometry

The PI, Christopher Bishop, will study problems in two and three dimensional geometry which come from classical complex analysis, the theory of quasiconformal mappings, hyperbolic geometry and computational geometry, with a particular emphasis on interactions between these areas. In recent work the PI has shown that ideas from hyperbolic and computational geometry could help resolve a question of classical analysis: how to efficiently compute conformal maps onto planar domains. This work will be extended to new settings, and will be used to study problems arising in computational geometry, e.g., efficient generation of meshes with desirable properties. The PI will investigate the possible application of the carpenter's rule problem from computational geometry to an analysis problem about deforming chord-arc curves. He will also continue his work on the distortion properties of quasiconformal mappings and on the conformal welding problem.The PI, Christopher Bishop, will continue his investigations into the theory and computation of conformal and quasiconformal maps. Conformal maps send one region to another so that angles (but not necessarily distances) are preserved. Quasiconformal maps may distort angles, but only by a limited amount. One of the oldest and most famous examples is the Mercator projection which maps the spherical Earth to flat piece of paper. This can't be done without some sort of distortion, and for navigation it is more convenient to have angles preserved than distances (so looking at a chart you know the correct direction to head, if not the exact distance to be covered). Conformal maps also appear in many applications where one wants to transfer a problem from a domain with complicated geometry to a simpler one (such as a disk) where it is easier to solve. Examples come from aerodynamics, fluid flow, vibrating membranes, heat flow, electrostatics and many other problems. Conformal maps also play a fundamental role within pure mathematics in areas such as dynamics, complex analysis, probability and geometry. Because of their importance, there are numerous techniques for computing conformal maps numerically, but different methods work best in different situations and often a method fails for sufficiently complicated regions. The PI has developed an algorithm that is guaranteed to work for a large class of regions, and is able to estimate the time needed to achieve a given accuracy. The algorithm is based on new connections between geometric concepts from computer science and non-Euclidean geometry arising in pure mathematics. Under the current proposal the PI will seek to turn this theoretical algorithm into a practical method, investigate generalizations more complicated regions and to higher dimensions, and apply the method to problems arising from computer science.

PI克里斯托弗·毕晓普（Christopher Bishop）将研究来自经典复杂分析，准文字映射，双曲线几何形状和计算几何学理论的两个和三维几何形状的问题，并特别强调了这些区域之间的相互作用。在最近的工作中，PI表明，来自双曲线和计算几何形状的思想可以帮助解决经典分析的问题：如何有效地将共形图计算到平面域上。 这项工作将扩展到新的设置，并将用于研究在计算几何形状中引起的问题，例如，具有理想特性的网格产生。 PI将研究木匠规则问题的可能应用，从计算几何形状到有关变形和弦曲线的分析问题。他还将继续他的工作在准表片映射的扭曲属性和保形焊接问题上。PIChristopher Bishop将继续他对保形和准文献图的理论和计算的调查。共形图将一个区域发送到另一个区域，以便保留角度（但不一定是距离）。准文化图可能会扭曲角度，但仅量有限。最古老，最著名的例子之一是墨托投影，它将球形地球映射到平坦的纸。如果没有某种失真，这是无法做到的，而导航的角度比距离保持更方便（因此，查看图表，您知道正确的方向，即使不是要覆盖的确切距离）。共形图也出现在许多应用程序中，其中一个人希望将问题从具有复杂几何形状的域转移到更简单的几何形状（例如磁盘）中，在该域更容易求解。例如来自空气动力学，流体流动，振动膜，热流，静电和许多其他问题。在动态，复杂分析，概率和几何形状等领域，共形图在纯数学中也起着基本作用。由于它们的重要性，有许多用于计算保形图的技术，但在不同情况下，不同的方法最有效，并且通常是一种方法，方法是对足够复杂的区域失败。 PI开发了一种算法，该算法可以保证为大量区域工作，并能够估算达到给定准确性所需的时间。 该算法基于计算机科学的几何概念与在纯数学中产生的非欧国人几何形状之间的新联系。根据当前的提案，PI将寻求将这种理论算法转变为一种实用方法，研究更复杂的区域和更高维度的概括，并将该方法应用于计算机科学引起的问题。