Accelerating visual communication of mathematical knot deformation
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R E G UL A R P A P E R
Juan Lin • Hui Zhang
Accelerating visual communication of mathematical knot deformation
Received: 29 January 2020 / Revised: 8 April 2020 / Accepted: 15 May 2020 Ó The Visualization Society of Japan 2020
Abstract Mathematical knots are different from everyday ropes, in that they are infinitely stretchy and flexible when being deformed into their ambient isotopic. For this reason, challenges of visualization and computation arise when communicating mathematical knot’s static and changing structures during its topological deformation. In this paper, we focus on visual and computational methods to facilitate the communication of mathematical knot’ dynamics by simulating the topological deformation and capturing the critical changes during the entire simulation. To improve our visual experience, we design and exploit parallel functional units to accelerate both topological refinements in simulation phase and view selection in presentation phase. To further allow a real-time keyframe-based communication of knot deformation, we propose a fast and adaptive method to extract key moments where only critical changes occur to represent and summarize the long deformation sequence in real-time fashion. We conduct performance study and present the efficacy and efficiency of our methods. Keywords Knot untanglement View selection Least squares fitting Parallelization
1 Introduction One of the fundamental problems in knot theory (Adams 2004) is to determine whether a closed mathematical curve can be deformed into a ring (or ‘‘unknot’’) without cutting or passing through the curve itself. The objects being studied, i.e., mathematical knots, are familiar and appear similar to the 3D ropes in our everyday life, except that the mathematical ones are infinitely stretchy and flexible during deformation to their topological equivalence. The mathematical way of untangling a knot is to perform Reidemeister moves (Trace 1983), which reduce all knot deformations to a sequence of three types of ‘‘moves’’ called the twist move, poke move, and slide move. In principle, knot untanglement can be presented as a sequence of such simple (and powerful) moves. However, choosing the right combination of the Reidemeister moves in the right order can be very challenging. The whole procedure is often error-prone and thus requires technical expertise. Our goal in this paper is to develop an interactive visual tool that requires minimal knot theory expertise to model and present mathematical knots that can evolve and untangle by themselves into simplified structures. We start with a family of interactive methods to sketch mathematical knots as closed node-link diagrams with ‘‘energy’’ charged at each node. In this way, mathematical knots untangle by themself from a higher energy state to the lower. The complete untanglement can take a fairly long sequence of deformation to simulate. We then proceed to develop an algorithm to capture the key visual frames from the entire simulation where
J. Lin H. Zhang (&) Computer Engine
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