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.

翻译：曲面的可展性可通过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。该主题的计算方法涵盖从特殊参数化到离散等距映射。然而，目前尚缺乏针对一般四边形网格的局部可展性判据。本文提出一种新颖且高效的离散可展性判据，适用于带顶点权重的四边形网格，其动机源于微分几何中一个著名的特征——即第二基本形式的秩亏缺。我们将接触元素分配给网格面，将直纹向量分配给边，组合后得到每个面的可展性条件。利用标准优化流程，我们能够实现交互式设计与可展放样。采用的网格是组合正则的四边网格，包含孤立奇异点，但无需遵循可展曲面上任何特殊曲线。因此，它们可轻松嵌入涉及重网格化、裁剪与合并等标准操作的设计工作流中。一个重要特性是：我们可直接从网格推导出防水、有理双二次样条曲面。值得注意的是，该曲面是加权Doo-Sabin细分（在接触元素上以插值方式作用）的极限。