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We present a computer interface to visualize and interact with mathematical knots, i.e., the embeddings of closed circles in 3-dimensional Euclidean space. Mathematical knots are slightly different than everyday knots in that they are infinitely stretchy and flexible when being deformed into their topological equivalence. In this work, we design a visualization interface to depict mathematical knots as closed node-link diagrams with energies charged at each node, so that highly-tangled knots can evolve by themselves from high-energy states to minimal (or lower) energy states. With a family of interactive methods and supplementary user interface elements, out tool allows one to sketch, edit, and experiment with mathematical knots, and observe their topological evolution towards optimal embeddings. In addition, out interface can extract from the entire knot evolution those key moments where successive terms in the sequence differ by critical change; this provides a clear and intuitive way to understand and trace mathematical evolution with a minimal number of visual frames. Finally out interface is adapted and extended to support the depiction of mathematical links and braids, whose mathematical concepts and interactions are just similar to our intuition about knots. All these combine to show a mathematically rich interface to help us explore and understand a family of fundamental geometric and topological problems.
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Visually Communicating Mathematical Knot Deformation
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, a number of challenges arise when visualizing mathematical knot's static and changing structures during topological deformation. In this paper we focus on computational methods to visually communicate the mathematical knot's dynamics by computationally simulating the topological deformation and capturing the critical changes during the entire simulation. To further improve our visual experience, we propose a fast and adaptive method to extract key moments where only critical changes occur to represent and summarize the long deformation sequence. We conduct evaluation study to showcase the efficacy and efficiency of our methods.
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- Award ID(s):
- 1651581
- PAR ID:
- 10140256
- Date Published:
- Journal Name:
- VINCI'2019: Proceedings of the 12th International Symposium on Visual Information Communication and Interaction
- Page Range / eLocation ID:
- 1 to 8
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3356422.3356438
