NYI: Surface Splines Over Irregular Meshes

制造中的压铸件模型水平集 功能，模拟中的沉积前沿，令人愉悦的形状 在动画、快速原型制作和虚拟环境中交互的可视化方面， 世界上有一个共同点：需要有效的表示， 光滑的表面。 目前，我们对科学的一些认识和使用 结果是由事实，即标准的代表性， 用于任务，张量积B样条，只能模拟一个小的子集， 这些表面自然和有效地，即表面， 规则的棋盘式排列。 自由曲面 样条扩展了样条范例和技术，以涵盖不规则 安排 在扩展中，它们保留了 通过用超平面切割的过程从控制网格中切割表面，还 称为切角或细分。 到目前为止只有一阶正切 存在连续的表面样条。 高阶连续性的扩展是 的根本和实际利益。 基本方面包括一个新的理论， 使用强大的计算能力， 这与在经典高斯微分几何中分析曲面不同。 本研究的目标是开发一个通用的框架，表面 样条曲线，并使用表示来优化曲面的形状， 考虑到各种成本函数。 直接目标是推导出 曲面样条曲线，并找到工具来调整变化 曲率是曲面的重要质量标准。 以来 挑战是相当大的比一阶曲面样条的定义 在不规则网格上，进展需要大量使用符号计算 和图形进行质量评估。

Models of die casts in manufacturing, level sets of functions, deposition fronts in simulation, pleasing shapes in animation, fast prototyping, and visualization of interactions in a virtual world have one thing in common: the need for efficient representations of smooth surfaces. At present, some of our understanding and use of scientific results is handicapped by the fact that the standard representation used for the task, tensor-product B-splines, can only model a small subset of these surfaces naturally and efficiently, namely surfaces that allow for a regular, checker-board arrangement of its pieces. Free-form surface splines extend the spline paradigm and techniques to cover also irregular arrangements. In extending, they preserve the intuitive generation of the surface from the control mesh by a process of cutting with hyperplanes, also known as corner cutting or subdivision. So far only first-order, tangent continuous surface splines exist. Extension to higher-order continuity is both of fundamental and practical interest. The fundamental aspect consists of a new theory that arises when surfaces from pieces using massive computing power in contrast to analyzing surfaces as in classical Gaussian differential geometry. The goal of this research is to develop a general framework for surface splines and to use the representation to optimize the shape of surfaces with respect to various cost functions. The immediate goal is the derivation of surface splines and to find tools to adjust the variation of curvature, an important quality criterion for surfaces. Since the challenge is considerably greater than the definition of first-order surface splines over irregular meshes, progress requires heavy use of symbolic computation and graphics for quality assessment.