论文标题
排斥曲线
Repulsive Curves
论文作者
论文摘要
曲线在计算机图形，物理模拟和数学可视化中起着基本作用，但是大多数用于曲线设计的工具无助于防止穿越或自我交流。本文开发了（自我）对平面和空间曲线的（自我）排斥的有效算法，这些算法非常适合计算设计中的问题。我们的起点是所谓的切线能量，它为自我干预提供了无限的障碍。与例如物理模拟中使用的局部碰撞检测策略相反，这种能量考虑了所有对点之间的相互作用，因此对于全球形状优化很有用：局部最小值倾向于在美学上是令人愉悦的，物理上有效的，并且在太空中分布得很好。基于sobolev-slobodeckij内部产品的梯度下降的重新制定使我们能够快速发展局部最小值---独立于曲线分辨率。我们还开发了一个分层多机构方案，该方案可大大降低每个步骤优化成本。能量很容易与各种约束和惩罚（例如，不可延伸性或避免障碍物）进行集成，我们将其用于应用程序，包括曲线堆积，缠结，无缠结，图形嵌入，非跨条带插值，流量插值，流动可视化和机器人路径计划。
Curves play a fundamental role across computer graphics, physical simulation, and mathematical visualization, yet most tools for curve design do nothing to prevent crossings or self-intersections. This paper develops efficient algorithms for (self-)repulsion of plane and space curves that are well-suited to problems in computational design. Our starting point is the so-called tangent-point energy, which provides an infinite barrier to self-intersection. In contrast to local collision detection strategies used in, e.g., physical simulation, this energy considers interactions between all pairs of points, and is hence useful for global shape optimization: local minima tend to be aesthetically pleasing, physically valid, and nicely distributed in space. A reformulation of gradient descent, based on a Sobolev-Slobodeckij inner product enables us to make rapid progress toward local minima---independent of curve resolution. We also develop a hierarchical multigrid scheme that significantly reduces the per-step cost of optimization. The energy is easily integrated with a variety of constraints and penalties (e.g., inextensibility, or obstacle avoidance), which we use for applications including curve packing, knot untangling, graph embedding, non-crossing spline interpolation, flow visualization, and robotic path planning.
