NYI: Surface Splines Over Irregular Meshes

制造中的压铸模型、功能的级别集、模拟中的沉积前沿、动画中的美观形状、快速原型以及虚拟世界中交互的可视化都有一个共同点：需要高效地表示光滑的表面。目前，我们对科学结果的一些理解和使用受到以下事实的限制：用于任务的标准表示，张量积B-样条线，只能自然和有效地模拟这些曲面的一小部分，即允许其棋子规则地排列的曲面。自由形式的曲面样条线扩展了样条线的范例和技术，以涵盖不规则排列。在延伸过程中，它们通过使用超平面进行切割的过程(也称为拐角切割或细分)来保留控制网格中曲面的直观生成。到目前为止，仅存在一阶切线连续曲面样条线。推广到高阶连续性具有重要的理论意义和实用价值。基本方面包括一个新的理论，当使用大量的计算能力将表面从碎片中分离出来时，与在经典的高斯微分几何中分析表面形成对比。这项研究的目的是开发一个通用的曲面样条线框架，并使用该表示法来优化相对于各种代价函数的曲面形状。目前的目标是导出曲面样条线，并找到调整曲率变化的工具，这是曲面的一个重要质量标准。由于这一挑战比不规则网格上一阶曲面样条线的定义要大得多，进展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.