High-quality Spline and Subdivision Surfaces

卓越的外形和表面质量对新产品设计和工业竞争力有着明显的影响。减少曲面建模时人工的启发式干预是精确的工程模拟和高质量的几何处理的前提，它减少了工作量、时间、成本和可变性。对于应用微分几何和几何设计来说，刻画和创建能够经受住详细审查的曲面是一个尚未解决的基本挑战。本研究将对高质量的曲面进行刻画，并开发用于创建曲面的工具。这项工作主要集中在计算效率高、有限和仿射不变的曲面方案上，并生成可用标准有理或多项式形式表示的光滑、曲率连续的参数曲面。最初的假设是，如果多个主曲面之间的混合不会产生定义数据中不存在的曲率特征，那么它是好的。研究将分析为什么现有技术无法生成令人满意的曲率分布，发展出实用的表面质量概念，修复现有技术或宣布它们为什么存在根本缺陷，并开发新的方法，以针对精心选择的多个主曲面连接的障碍路线进行测试。

Superior shape and surface quality have a visible impact on new product design and industrial competitiveness. Decreasing the need for human, heuristic intervention when modeling surfaces is a precondition for accurate engineering simulation and high-quality geometry processing and it reduces effort, time, cost and variability. Characterizing and creating surfaces that withstand scrutiny in the detail is a fundamental unresolved challenge to applied differential geometry and geometric design.This research will characterize high-quality surfaces and develop tools for their creation. The effort centers on surfacing schemes that are computationally efficient, finite and affine invariant, and generate fair, curvature-continuous, parametric surfaces that are representable in a standard rational or polynomial form. The starting hypothesis is that a blend between multiple primary surfaces is good if it does not create curvature features that were not present in the defining data.The research will analyze why the existing techniques fail to generate satisfactory curvature distributions, develop a practical notion of surface quality, repair existing techniques or declare why they are fundamentally flawed and develop new approaches to be tested against a carefully chosen obstacle course of joins of multiple primary surfaces.