A Geometric Approach to Simulating Knotting and Entanglement of Slender Objects

打结和缠结在我们的日常生活中扮演着至关重要的角色（头发、编织、航海、攀岩、手术缝合线、口袋耳机线、鞋带）。结在科学（DNA，分子）和工程（聚合物，纺织品，航海/航空航天）中也是至关重要的。尽管它们无处不在，也很重要，但我们对纠缠的基本理解仍处于起步阶段，即使是对于最基本的问题：(1)这个结能坚持住吗？（ii）是什么机制使这个结在一种情况下如此有效，而在另一种情况下却不有效？（三）哪些因素最有可能改变其功效？（iv）在什么情况下会自发形成某些结（口袋里的耳机线）或松开（跑步时的鞋带），如何防止这种情况发生？理解这些重要问题的一个途径是计算运动。使用计算机，我们可以预测材料如何移动或改变形状，因为他们的环境。当计算机预测材料如何移动和变形时，它们帮助我们理解、预测和更安全地与物理世界互动。从翻腾的头发到游动的微生物再到机器人，计算运动广泛地影响着艺术、科学和工程。该研究项目将开发计算打结和缠结运动的计算方法。该项目推进了这个问题的许多方面，从理论到应用。这项研究将解决以下问题：从数学上讲，电缆缠绕意味着什么？我们能否设计出计算机程序，保证在打结的场景中产生值得信赖的结果，而且速度快？如果在密集的纠缠网络中有大量的光纤，我们如何在保持预测结果质量的同时加速计算？这个项目将帮助将这些软件工具转移到加拿大的视觉特效和游戏行业，这样虚拟角色就可以拥有各种类型、形状、粗糙度和密度的逼真头发。我们将能够更好地模仿社会上各种各样的发型和类型。这项研究还将使用由此产生的计算机软件来研究有关纠缠的基本问题。例如，拿几千个回形针，把它们都弯成开瓶器，把它们放在一个封闭的盒子里，然后摇晃：我们预计会有一团乱麻。试着把回形针弯成直线：我们不再指望它们粘在一起了。那么其他形状呢，比如字母S、N或G呢？给定一根弯曲的电线，我们能预测它的自纠缠行为吗？这是一个基本的开放问题，对新材料的设计，对生物学的理解，甚至对工程“可编程物质”都有重要的影响。这个项目制作的软件将免费提供，任何人都可以在此基础上发展艺术、科学和工程。

Knotting and entanglement play a pivotal role in our daily lives (locs of hair, knitting, sailing, rockface climbing, surgical sutures, pocketed headphone cables, shoelaces). Knots are also pivotal in science (DNA, molecules) and engineering (polymers, textiles, nautical/aerospace). Despite their ubiquity and importance, our fundamental understanding of entanglement remains at its infancy, even for the most basic questions: (i) will this knot hold? (ii) what mechanism makes this knot so effective in one situation but not another? (iii) what factors are most likely to change its efficacy? (iv) under what conditions do certain knots spontaneously form (headphone cables in our pocket) or become undone (our shoelaces while running), and how can this be prevented? One avenue to understanding these important questions is to compute motion. Using a computer, we can predict how materials move or change shape due to their environment. When computers predict how materials move and deform, they help us to understand, predict, and more safely interact with the physical world. From billowing hair to swimming microorganisms to robotics, computing motion broadly impacts the arts, science and engineering. This research project will develop computational methods for computing the motions involved in knotting and entanglement. The project advances many aspects of this problem, from theoretical to applied. The research will address questions such as, what does it mean for a cable to become entangled, mathematically speaking? Can we design computer programs that are guaranteed to produce a trustworthy result for a knotting scenario, and do it fast? And if there are multitudes of fibres in a dense entangled network, how can we accelerate the computation while maintaining the quality of the resulting prediction? This project will help transfer these software tools to the Canadian visual special effects and games industries so that virtual characters can have realistic hair of all types, shapes, coarseness, and density. We will be better able to model the wide variety of hairstyles and types found in society. This research will also use the resulting computer software to study fundamental questions about entanglement. For instance, take a few thousand paperclips, bend them all into corkscrews, put them in a closed box, and shake: we expect a tangled mess. Try instead bending the paperclips into straight lines: we no longer expect them to cling to each other. What about for other shapes, such as the letters S, N, or G? Given a bent wire, can we predict its self-entanglement behaviour? This is a fundamental open question with significant ramifications for the design of new materials, understanding biology, and even engineering "programmable matter." The software produced by this project will be made freely available, enabling anyone to build on it to advance the arts, science and engineering.