A Geometric Approach to Simulating Knotting and Entanglement of Slender Objects

打结和缠绕在我们的日常生活中扮演着关键的角色(头发、编织、航海、攀岩、外科缝合、口袋耳机电缆、鞋带)。结在科学(DNA、分子)和工程(聚合物、纺织品、航海/航空航天)中也是关键。尽管它们无处不在，也很重要，但我们对纠缠的基本理解仍然处于初级阶段，甚至对于最基本的问题也是如此：(I)这个结还能打得住吗？(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是？(3)哪些因素最有可能改变其疗效？(Iv)在什么情况下，某些结会自发形成(口袋里的耳机线)或松开(跑步时鞋带)，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因其环境而移动或改变形状。当计算机预测材料如何移动和变形时，它们帮助我们理解、预测并更安全地与物理世界交互。从卷曲的头发到游泳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。这项研究项目将开发计算打结和纠缠所涉及的运动的计算方法。该项目从理论到应用对这一问题提出了许多方面的改进。这项研究将解决这样的问题，例如，从数学上讲，电缆缠绕意味着什么？我们设计的计算机程序能保证在打结的情况下产生可靠的结果，而且速度快吗？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快计算速度？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，以便虚拟角色可以拥有各种类型、形状、粗糙和密度的逼真头发。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关于纠缠的基本问题。例如，拿出几千个回形针，把它们都折成开塞螺丝钉，放在一个封闭的盒子里，摇晃：我们预计会出现一团乱麻。试着把回形针弯曲成直线：我们不再指望它们相互粘在一起。其他的形状呢，比如字母S、N或G？给出一根弯曲的导线，我们能预测它的自纠缠行为吗？这是一个根本性的悬而未决的问题，对新材料的设计、对生物学的理解，甚至对工程“可编程物质”都有重大影响。这个项目制作的软件将免费提供，使任何人都可以在它的基础上发展艺术、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.