Computational Geometry in the Physical World

随着计算机在物理世界中的应用越来越广泛，人们越来越需要计算工具来处理物体的物理属性，比如它们的几何形状。在过去的二十年中，计算几何领域取得了令人印象深刻的进步，为解决计算机图形学、计算机视觉、机器人和制造以及许多其他应用领域的许多几何问题提供了算法基础。本提案的目的是开发一套新颖的几何表示和算法技术，使计算几何在建模物理世界的对象和过程中更有用。其目的不是获得通用的、独立于域的解决方案——比如几何计算中鲁棒性问题的完整解决方案。相反，重点将放在某些关键问题上，如比例敏感性、形状可变性和运动，这些问题在物理对象和系统建模中几何的许多重要用途中出现。几何模型，适当地捕捉某些关键方面的不确定性或可变性的物理对象和过程的模型将被调查，并有效地操纵这些模型的有效算法将被开发。预计这组研究方向将产生新的数学和算法问题，这些问题将是有趣的，值得研究的，除了产生适合物理世界应用的几何工具之外。＊＊＊

9910633 Guibas, LeonidasAs the application of computers to the physical world widen, there is an ever increasing need for computational tools to deal with the physical attributes of objects, such as their geometry. The field of Computational Geometry has made impressive strides in the last twenty years in providing an algorithmic foundation for the solution to many geometric problems in computer graphics, computer vision, robotics and manufacturing, and many other application areas. The purpose of this proposal is to develop a set of novel geometric representations and algorithmic techniques that will make computational geometry more useful in modeling physical-world objects and processes. The aim is not to obtain universal, domain-independent solutions --- such as a complete solution to robustness concerns in geometric computing. Instead the focus will be on certain key issues, such as scale- sensitivity, shape variability, and motion, that arise in many of the important uses of geometry in modeling physical objects and systems. Geometric models that properly capture certain key aspects of the uncertainty or variability in models of physical objects and processes will be investigated, and efficient algorithms for manipulating these models effectively will be developed. It is expected that this set of research directions will generate new mathematical and algorithmic problems that will be interesting and worthy of study on their own right, in addition to producing geometric tools that are well-tailored to physical world applications. ***