Research Initiation Award: Towards the Effective Use of Multiple Data Structures for Geometric Problems

本研究将探讨用于表示曲面的两种不同数据结构（参数和代数）之间的关系，这两种数据结构已被用于解决同一组几何问题。目标几何问题包括离散数据插值问题和混合曲面的构造问题。将开发实用和建设性的算法，用于将切线连续参数曲面块的网格转换为切线连续代数曲面块的类似网格，反之亦然。在参数表示中影响曲面形状的有用参数与代数表示中影响曲面形状的有用参数之间的关系将通过理论和实验手段进行探讨。所采用的方法包括精确符号计算和数值逼近相结合。研究结果将用于构建新的解决方案，并改进现有的解决方案，以解决混合曲面的构建和流体流动的可视化问题。

This research will investigate the relationship between two different data structures for representing surfaces, parametric and algebraic, which have been used to solve the same set of geometric problems. The targeted geometric problems include the problem of scattered data interpolation and the problem of the construction of blending surfaces. Practical and constructive algorithms will be developed for converting a mesh of tangent-continuous parametric surface patches to an analogous mesh of tangent-continuous algebraic surface patches and vice-versa. Relationships between useful parameters influencing the shape of surfaces in the parametric representation to those in the algebraic representation will be explored through theoretical and experimental means. The methods to be employed include a combination of exact symbolic computation and numerical approximation. The results will be applied to construct new solutions and improve existing solutions to the problem of construction of blending surfaces and the visualization of fluid flow.