Computational Topology for Surface Approximation

这项研究探讨了广泛的问题，保证一个对象的基本形状是保留在工程设计，动画和分子建模中使用的算法。 虽然两个物体的几何特征，如表面积和体积，可以客观地测量和比较，但其他形状特征，如花园软管和打结的同一软管之间的区别，则更加微妙，需要本研究的方法。 令人惊讶的是，这些区别对于当今的算法来说并不容易，并且在检测这些差异方面的困难估计每年在航空和汽车工业中损失数十亿美元的生产力。 这项研究是计算机科学家和数学家之间的跨学科调查，相应的教育重点是向下一代科学家介绍一个整体的观点，将理论与新颖的工程和生命科学应用相结合。 分子建模的应用程序，预计将吸引许多妇女在生物科学，从而增加妇女在计算和数学science.This研究的参与新的拓扑约束和数值误差界的几何近似算法，定义当一个表面近似是在相同的拓扑等价类的原始表面。 虽然曲面近似是一个经典的数学主题，但拓扑等价的问题通常留给人类检查。 本研究的新方法是理论和实践的发展，以创建计算上易于处理的算法，确保拓扑等价的一个丰富的表面近似技术类。 创建适当的应用数学，以消除任何直接的算法依赖于中轴的计算显着丰富了一类强大的近似算法，提供可验证的拓扑结构。 另一个主要的创新是综合考虑几何模型组成的有界曲面片。 这种观点消除了一个流行的，但不切实际的理论假设，几何模型只有一个单一的边界表面。 额外的微妙之处依赖于新的数值近似沿着每个组成补丁的边界。

This research examines broad issues of guaranteeing that the essential shape of an object is preserved during algorithms used in engineering design, animation and molecular modeling. While geometric characteristics such as surface area and volume can be objectively measured and compared for two objects, other shape characterizations, such as the distinction between a garden hose and the same hose tied into a knot, are more subtle and require the methods of this research. Surprisingly, these distinctions are not easy for present day algorithms and difficulties in detecting these differences has been estimated to cost billions of dollars annually in lost productivity in the aeronautical and automotive industries. This research is an interdisciplinary investigation between computer scientists and mathematicians with a corresponding educational emphasis upon introducing the next generation of scientists to a holistic perspective that combines theory with novel engineering and life science applications. The applications to molecular modeling are expected to be attractive to the many women in the biological sciences and thereby increase the participation of women in the computational and mathematical sciences.This research integrates new topological constraints and numerical error bounds into geometric approximation algorithms, defining when a surface approximation is in the same topological equivalence class as the original surface. While surface approximation is a classical mathematical topic, the issue of topological equivalence has typically been left to human inspection. The novel approach of this research is the development of the theory and practice to create computationally tractable algorithms that ensure topological equivalence for a rich class of surface approximation techniques. The creation of appropriate applied mathematics to eliminate any direct algorithmic dependence upon computation of the medial axis significantly enriches the class of robust approximation algorithms delivering verifiable topology. Another major innovation is the comprehensive consideration of geometric models composed from bounded surface patches. This perspective eliminates a prevailing, but unrealistic theoretical hypothesis that geometric models have only a single bounding surface. The additional subtleties rely upon new numerical approximations along the boundaries of each constituent patch.