可展四边形网格与接触元素网
Developable Quad Meshes and Contact Element Nets
AI总结：
针对一般四边形网格缺少局部可展性判据的问题，提出基于面接触元素与边直纹向量的加权离散判据，支持交互式可展设计并导出无间隙有理双二次样条曲面。
摘要：
曲面的可展性质可以用不同的等价方式表达，例如高斯曲率为零，或存在到平面区域的等距映射。对该主题的计算性贡献涵盖从特殊参数化到离散等距映射等多种方法。然而，迄今为止一直缺乏一种用于表达一般四边形网格可展性的局部判据。本文提出了一种新颖且高效的离散可展性判据，该判据应用于带有顶点权重的四边形网格，其动机来自微分几何中一个著名的刻画，即秩亏的第二基本形式。我们为网格的面分配接触元素，并为边分配直纹向量，二者结合给出每个面的可展性条件。利用标准优化过程，我们能够进行交互式设计和可展放样。我们采用的网格是具有孤立奇点的组合规则四边形网格，除此之外不要求其遵循可展曲面上的任何特殊曲线。因此，它们很容易嵌入涉及重网格化、修剪和合并等标准操作的设计工作流中。一个重要特性是，我们可以从网格直接导出无间隙的有理双二次样条曲面。值得注意的是，该曲面作为加权 Doo-Sabin 细分的极限出现，而这种细分以插值方式作用于接触元素。
英文摘要：
The property of a surface being developable can be expressed in different equivalent ways, by vanishing Gauss curvature, or by the existence of isometric mappings to planar domains. Computational contributions to this topic range from special parametrizations to discrete-isometric mappings. However, so far a local criterion expressing developability of general quad meshes has been lacking. In this paper, we propose a new and efficient discrete developability criterion that is applied to quad meshes equipped with vertex weights, and which is motivated by a well-known characterization in differential geometry, namely a rank-deficient second fundamental form. We assign contact elements to the faces of meshes and ruling vectors to the edges, which in combination yield a developability condition per face. Using standard optimization procedures, we are able to perform interactive design and developable lofting. The meshes we employ are combinatorial regular quad meshes with isolated singularities but are otherwise not required to follow any special curves on a developable surface. They are thus easily embedded into a design workflow involving standard operations like remeshing, trimming, and merging operations. An important feature is that we can directly derive a watertight, rational bi-quadratic spline surface from our meshes. Remarkably, it occurs as the limit of weighted Doo-Sabin subdivision, which acts in an interpolatory manner on contact elements.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
