Computation and visualization of ideal knot shapes
Computation and visualization of ideal knot shapes
复制标题
理想结形状的计算和可视化
DOI：
10.5075/epfl-thesis-4621
复制
发表时间：
2010
影响因子：
5.2
通讯作者：
M. Carlen
中科院分区：
文献类型：
--
作者：
M. Carlen
我们研究了在一根绳子上打结问题的数值模拟和可视化。目标是使用尽可能少的固定规定半径的绳子来打特定的结，例如绳结。三叶形、八字形等等。结的绳索长度（要最小化的比率）是其长度除以半径。给出了最小化绳索长度的现有算法的概述。它们基于不同的离散化。我们的工作建立在 biarc 离散化的基础上，为此我们开发了整个 C++ 库“libbiarc”。该库包含各种用于操作曲线、结或链接的工具。双弧离散化特别适合厚度评估。为了计算理想的结形状，我们使用模拟退火软件，该软件也包含在“libbiarc”中，用于双弧离散化。模拟退火是一种随机优化算法，随机改变点或切线数据。在寻找适合该过程的移动过程中，我们找到了结的傅里叶表示，它允许在退火过程中对曲线进行全局变化。此外，通过傅里叶表示，我们可以强制给定的结可能具有的对称性。为了识别这些对称性，我们使用不强制对称性的模拟可视化。结形状及其属性的可视化是这项工作的另一个重要方面。它的范围从简单的结曲率图，到特定距离、圆或球函数的二维图，再到接触特性的 3 维图像。开发了专门设计的颜色渐变来强调绘图的关键区域。我们表明，理想环面结的接触集是一条与结本身同位素的曲线，这是通过可视化首次提出的结果。数字和可视化的结合让我们意识到三叶结（9 号台球）内的闭合轨迹。因此，对称性和台球使得仅用两个曲线子段来表示三叶形成为可能。我们还对单位 3 球体或 S3 中的结形状进行退火和可视化。特别是，我们提出了候选最优性的接触集，其弯曲接触弦形成 Villarceau 圆，而 Villarceau 圆又跨越嵌入单位 3 球体中的 Clifford 环面。最后，使用 3D 打印机将一些结和接触面构建为物理 3D 模型。
We investigate numerical simulations and visualizations of the problem of tying a knot in a piece of rope. The goal is to use the least possible rope of a fixed, prescribed radius to tie a particular knot, e.g. a trefoil, a figure eight, and so on. The ropelength of the knot, the ratio to be minimized, is its length divided by its radius. An overview of existing algorithms to minimize the ropelength is given. They are based on different discretizations. Our work builds on the biarc discretization, for which we have developed an entire C++ library "libbiarc". The library contains a variety of tools to manipulate curves, knots or links. The biarc discretization is particularly well suited to evaluation of thickness. To compute ideal knot shapes we use simulated annealing software, which is also included in "libbiarc", on a biarc discretization. Simulated annealing is a stochastic optimization algorithm that randomly changes the point or tangent data. In the quest to find appropriate moves for this process we arrived upon a Fourier representation for knots, which allows global changes to the curve in the annealing process. Moreover, with the Fourier representation we can enforce symmetries that a given knot might have. To identify these symmetries we use visualization of simulations where symmetry was not enforced. Visualization of knot shapes and their properties is another important aspect in this work. It ranges from simple graphs of the curvature of a knot, through 2-dimensional plots of certain distance, circle or sphere functions, to 3-dimensional images of contact properties. Specially designed color gradients have been developed to emphasize crucial regions of the plots. We show that the contact set of ideal torus knots is a curve that is ambient isotopic to the knot itself, which is a result first suggested by visualization. A combination of numerics and visualization made us aware of a closed trajectory within the trefoil knot, a 9-billiard. Consequently the symmetries and the billiard make it possible to represent the trefoil with only two curve sub segments. We also anneal and visualize knot shapes in the unit 3-sphere or S3. In particular we present the contact set of a candidate for optimality, whose curved contact chords form Villarceau circles, which in turn span a Clifford torus embedded in the unit 3-sphere. Finally some knots and contact surfaces are constructed as physical 3D models using 3D printers.