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Fast Linking Numbers for Topology Verification of Loopy Structures
arXiv - CS - Graphics Pub Date : 2021-06-23 , DOI: arxiv-2106.12655
Ante Qu, Doug L. James

It is increasingly common to model, simulate, and process complex materials based on loopy structures, such as in yarn-level cloth garments, which possess topological constraints between inter-looping curves. While the input model may satisfy specific topological linkages between pairs of closed loops, subsequent processing may violate those topological conditions. In this paper, we explore a family of methods for efficiently computing and verifying linking numbers between closed curves, and apply these to applications in geometry processing, animation, and simulation, so as to verify that topological invariants are preserved during and after processing of the input models. Our method has three stages: (1) we identify potentially interacting loop-loop pairs, then (2) carefully discretize each loop's spline curves into line segments so as to enable (3) efficient linking number evaluation using accelerated kernels based on either counting projected segment-segment crossings, or by evaluating the Gauss linking integral using direct or fast summation methods (Barnes-Hut or fast multipole methods). We evaluate CPU and GPU implementations of these methods on a suite of test problems, including yarn-level cloth and chainmail, that involve significant processing: physics-based relaxation and animation, user-modeled deformations, curve compression and reparameterization. We show that topology errors can be efficiently identified to enable more robust processing of loopy structures.

中文翻译:

用于环状结构拓扑验证的快速链接数

基于环状结构对复杂材料进行建模、模拟和处理越来越普遍,例如在纱线级布料服装中,这些材料在环间曲线之间具有拓扑约束。虽然输入模型可能满足闭环对之间的特定拓扑联系,但后续处理可能会违反这些拓扑条件。在本文中,我们探索了一系列有效计算和验证闭合曲线之间的连接数的方法,并将这些方法应用于几何处理、动画和模拟中,以验证在处理过程中和处理后保留拓扑不变量。输入模型。我们的方法分为三个阶段:(1)我们识别潜在的交互循环对,然后(2)仔细离散每个循环' s 样条曲线成线段,以便 (3) 使用加速内核进行有效的链接数评估,基于计算投影的线段交叉,或通过使用直接或快速求和方法(Barnes-Hut 或快速多极子)评估高斯链接积分方法)。我们在一系列测试问题上评估这些方法的 CPU 和 GPU 实现,包括纱线级布料和链甲,这些问题涉及重要的处理:基于物理的松弛和动画、用户建模的变形、曲线压缩和重新参数化。我们表明可以有效地识别拓扑错误,以便对循环结构进行更稳健的处理。或通过使用直接或快速求和方法(Barnes-Hut 或快速多极方法)评估高斯链接积分。我们在一系列测试问题上评估这些方法的 CPU 和 GPU 实现,包括纱线级布料和链甲,这些问题涉及重要的处理:基于物理的松弛和动画、用户建模的变形、曲线压缩和重新参数化。我们表明可以有效地识别拓扑错误,以便对循环结构进行更稳健的处理。或通过使用直接或快速求和方法(Barnes-Hut 或快速多极方法)评估高斯链接积分。我们在一系列测试问题上评估这些方法的 CPU 和 GPU 实现,包括纱线级布料和链甲,这些问题涉及重要的处理:基于物理的松弛和动画、用户建模的变形、曲线压缩和重新参数化。我们表明可以有效地识别拓扑错误,以便对循环结构进行更稳健的处理。曲线压缩和重新参数化。我们表明可以有效地识别拓扑错误,以便对循环结构进行更稳健的处理。曲线压缩和重新参数化。我们表明可以有效地识别拓扑错误,以实现对循环结构的更稳健处理。
更新日期:2021-06-25
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