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，对处理对象的物理属性（例如其几何形状）的计算工具的需求也越来越大。 计算几何领域在过去的二十年里取得了令人印象深刻的进步，为计算机图形学，计算机视觉，机器人和制造业以及许多其他应用领域中的许多几何问题的解决方案提供了算法基础。这个提议的目的是开发一套新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。我们的目标不是获得普遍的，域独立的解决方案-如一个完整的解决方案，在几何计算的鲁棒性问题。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，这些问题出现在许多重要的几何建模物理对象和系统中。几何模型，适当地捕捉某些关键方面的不确定性或物理对象和过程的模型的可变性将进行调查，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，这些问题本身将是有趣的，值得研究的，除了产生适合物理世界应用的几何工具。 ***

As 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. ***