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Issue No.12 - Dec. (2012 vol.18)
pp: 2051-2060
Hui Zhang , Indiana University
Jianguang Weng , Zhejiang University of Media and Communications, China
Lin Jing , Fujian Agriculture and Forestry University, Fuzhou, China
Yiwen Zhong , Fujian Agriculture and Forestry University, Fuzhou, China
ABSTRACT
We present KnotPad, an interactive paper-like system for visualizing and exploring mathematical knots; we exploit topological drawing and math-aware deformation methods in particular to enable and enrich our interactions with knot diagrams. Whereas most previous efforts typically employ physically based modeling to simulate the 3D dynamics of knots and ropes, our tool offers a Reidemeister move based interactive environment that is much closer to the topological problems being solved in knot theory, yet without interfering with the traditional advantages of paper-based analysis and manipulation of knot diagrams. Drawing knot diagrams with many crossings and producing their equivalent is quite challenging and error-prone. KnotPad can restrict user manipulations to the three types of Reidemeister moves, resulting in a more fluid yet mathematically correct user experience with knots. For our principal test case of mathematical knots, KnotPad permits us to draw and edit their diagrams empowered by a family of interactive techniques. Furthermore, we exploit supplementary interface elements to enrich the user experiences. For example, KnotPad allows one to pull and drag on knot diagrams to produce mathematically valid moves. Navigation enhancements in KnotPad provide still further improvement: by remembering and displaying the sequence of valid moves applied during the entire interaction, KnotPad allows a much cleaner exploratory interface for the user to analyze and study knot equivalence. All these methods combine to reveal the complex spatial relationships of knot diagrams with a mathematically true and rich user experience.
INDEX TERMS
Visualization, Mice, Rendering (computer graphics), Shape, Haptic interfaces, Media, User interfaces, Math Visualization, Knot Theory
CITATION
Hui Zhang, Jianguang Weng, Lin Jing, Yiwen Zhong, "KnotPad: Visualizing and Exploring Knot Theory with Fluid Reidemeister Moves", IEEE Transactions on Visualization & Computer Graphics, vol.18, no. 12, pp. 2051-2060, Dec. 2012, doi:10.1109/TVCG.2012.242
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