Variational Surface Cutting
Variational Surface Cutting
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DOI：
10.1145/3197517.3201356
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发表时间：
2018-08-01
影响因子：
6.2
通讯作者：
Crane, Keenan
中科院分区：
文献类型：
--
作者：
Sharp, Nicholas;Crane, Keenan
本文提出了一种整体变分法来切割曲面，使曲面能以低度量变形被展平到平面上。这种切割是各种算法中的关键组成部分，这些算法寻求在平坦域上参数化表面，或从平坦材料制造结构。而不是仅仅基于曲线本身的属性（例如，它的长度或曲率），我们制定了一个流程，直接优化切割和平整引起的变形。值得注意的是，我们不必显式地参数化曲面以评估切割的成本，而是可以集成在切割曲线本身上定义的简单演化方程。我们到达这个流动通过一个新的应用程序的形状衍生物的Yamabe方程从共形几何。然后，我们开发了一个欧拉数值积分三角化的表面，它不限制削减网格边缘，并可以将用户定义的数据，如重要性或闭塞。由此产生的切割曲线可用于驱动失真到任意低的水平，并具有非常不同的字符从通过纯离散公式获得的切割。我们简要地探讨了计算设计的潜在应用，以及连接到空间填充曲线和均匀热分布的问题。
This paper develops a global variational approach to cutting curved surfaces so that they can be flattened into the plane with low metric distortion. Such cuts are a critical component in a variety of algorithms that seek to parameterize surfaces over flat domains, or fabricate structures from flat materials. Rather than evaluate the quality of a cut solely based on properties of the curve itself (e.g., its length or curvature), we formulate a flow that directly optimizes the distortion induced by cutting and flattening. Notably, we do not have to explicitly parameterize the surface in order to evaluate the cost of a cut, but can instead integrate a simple evolution equation defined on the cut curve itself. We arrive at this flow via a novel application of shape derivatives to the Yamabe equation from conformal geometry. We then develop an Eulerian numerical integrator on triangulated surfaces, which does not restrict cuts to mesh edges and can incorporate user-defined data such as importance or occlusion. The resulting cut curves can be used to drive distortion to arbitrarily low levels, and have a very different character from cuts obtained via purely discrete formulations. We briefly explore potential applications to computational design, as well as connections to space filling curves and the problem of uniform heat distribution.