NYI: Surface Splines Over Irregular Meshes

制造中的压铸模型、功能水平集、仿真中的沉积前沿、动画中的令人愉悦的形状、快速原型和虚拟世界中交互的可视化有一个共同点：对光滑表面的有效表示的需求。目前，我们对科学结果的一些理解和使用受到这样一个事实的阻碍，即用于任务的标准表示，张量积b样条，只能自然有效地模拟这些表面的一小部分，即允许其碎片有规则的棋盘排列的表面。自由曲面样条扩展了样条范式和技术，以覆盖不规则的排列。在扩展中，它们通过使用超平面切割（也称为角切割或细分）的过程，保持了从控制网格直观生成表面的能力。到目前为止，只有一阶切线连续曲面样条存在。向高阶连续性的推广具有重要的基础意义和实际意义。基本方面包括一个新的理论，当从碎片中提取曲面时，使用大量的计算能力，而不是像经典高斯微分几何那样分析曲面。本研究的目标是为曲面样条开发一个通用框架，并使用该表示根据各种成本函数来优化曲面的形状。直接的目标是曲面样条的推导和找到工具来调整曲率的变化，曲率是曲面的一个重要的质量标准。由于挑战比不规则网格上的一阶曲面样条的定义要大得多，因此需要大量使用符号计算和图形来进行质量评估。

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.