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Accelerating visual communication of mathematical knot deformation

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.

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References

  • Adams CC (2004) The knot book: an elementary introduction to the mathematical theory of knots. American Mathematical Society, Providence

    MATH  Google Scholar 

  • Cao W, Hu P, Li H, Lin Z (2010) Canonical viewpoint selection based on distance-histogram. J Comput Aided Des Comput Gr 22(9):1515–1521

    Google Scholar 

  • Carlen M (2010) Computation and visualization of ideal knot shapes. Tech. rep, EPFL

  • Cashbaugh J, Kitts C (2018) Automatic calculation of a transformation matrix between two frames. IEEE Access 6:9614–9622

    Article  Google Scholar 

  • Chen Y, Guan Z, Zhang R, Du X, Wang Y (2019) A survey on visualization approaches for exploring association relationships in graph data. J Vis 22(3):625–639

    Article  Google Scholar 

  • Colin C (1988) A system for exploring the universe of polyhedral shapes. In: Eurographics 88

  • Costagliola G, De Rosa M, Fish A, Fuccella V, Saleh R, Swartwood S (2016) Knotsketch: a tool for knot diagram sketching, encoding and re-generation. J Vis Lang Sentient Syst 2:16–25

    Google Scholar 

  • Eades P (1984) A heuristic for graph drawing. Congr Numer 42:149–160

    MathSciNet  Google Scholar 

  • Erten C, Harding PJ, Kobourov SG, Wampler K, Yee G (2004) Exploring the computing literature using temporal graph visualization. In: Visualization and data analysis 2004, vol 5295. International Society for Optics and Photonics, pp 45–56

  • Garey MR, Johnson DS (1983) Crossing number is np-complete. SIAM J Algebraic Discrete Methods 4(3):312–316

    MathSciNet  Article  Google Scholar 

  • Harel D, Koren Y (2000) A fast multi-scale method for drawing large graphs. In: International symposium on graph drawing. Springer, pp 183–196

  • Johan H, Li B, Wei Y et al (2011) 3d model alignment based on minimum projection area. Vis Comput 27(6–8):565

    Article  Google Scholar 

  • Kobourov SG (2012) Spring embedders and force directed graph drawing algorithms. arXiv preprint arXiv:1201.3011

  • Lee CH, Varshney A, Jacobs DW (2005) Mesh saliency. In: ACM transactions on graphics (TOG), vol. 24. ACM, pp 659–666

  • Lin J, Hui Z (2019) Visualizing mathematical knot equivalence. In: IS&T international symposium on electronic imaging 2019, Visualization and Data Analysis 2019 proceedings, VDA 2019. Society for Imaging Science and Technology

  • Lin J, Zhang H (2019) Visually communicating mathematical knot deformation. In: Proceedings of the 12th international symposium on visual information communication and interaction. Association for Computing Machinery, New York, NY, USA. https://doi.org/10.1145/3356422.3356438

  • Liu L, Silver D, Bemis K (2019) Visualizing events in time-varying scientific data. J Vis 5:1–16

    Google Scholar 

  • Liu YJ, Fu QF, Liu Y, Fu XL (2012) 2d-line-drawing-based 3d object recognition. In: International conference on computational visual media. Springer, pp 146–153

  • Openmp. http://www.openmp.org/. Accessed Jan 7, 2020

  • Pach J, Tardos G (2002) Untangling a polygon. In: Mutzel P, Jünger M, Leipert S (eds) Graph drawing. Springer, Berlin, pp 154–161

    Chapter  Google Scholar 

  • Penrose R (1955) A generalized inverse for matrices. In: Mathematical proceedings of the Cambridge philosophical society, vol. 51. Cambridge University Press, pp 406–413

  • Schaefer M (2013) The graph crossing number and its variants: a survey. Electron J Combin 1000:21–22

    Article  Google Scholar 

  • Scharein RG (1998) Interactive topological drawing. Ph.D. thesis, Citeseer

  • Simon JK (1994) Energy functions for polygonal knots. J Knot Theory Ramif 3(03):299–320

    MathSciNet  Article  Google Scholar 

  • Snibbe S, Anderson S, Verplank B (1998) Springs and constraints for 3d drawing. In: Proceedings of the third phantom users group workshop

  • Trace B (1983) On the reidemeister moves of a classical knot. In: Proceedings of the American Mathematical Society, pp 722–724

  • Vázquez PP, Feixas M, Sbert M, Heidrich W (2001) Viewpoint selection using viewpoint entropy. VMV 1:273–280

    Google Scholar 

  • Vázquez PP, Sbert M (2003) Fast adaptive selection of best views. In: International conference on computational science and its applications. Springer, pp 295–305

  • Wu Y (1996) An md knot energy minimizing program. Department of Mathematics, University of Iowa

  • Zhang H, Thakur S, Hanson AJ (2007) Haptic exploration of mathematical knots. In: International symposium on visual computing. Springer, pp 745–756

  • Zhang H, Weng J, Jing L, Zhong Y (2012) Knotpad: visualizing and exploring knot theory with fluid reidemeister moves. IEEE Trans Vis Comput Gr 18(12):2051–2060

    Article  Google Scholar 

  • Zhang H, Weng J, Ruan G (2014) Visualizing 2-dimensional manifolds with curve handles in 4d. IEEE Trans Vis Comput Gr 1:1–1

    Google Scholar 

  • Zhang H, Zhong Y, Jiang J (2016) Visualizing knots and braids with touchable 3d manipulatives. In: 2016 IEEE Pacific visualization symposium (PacificVis), pp 24–31. https://doi.org/10.1109/PACIFICVIS.2016.7465247

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Acknowledgements

This work was supported in part by National Science Foundation Grant #1651581 and the 2016 ORAU’s Ralph E. Powe Junior Faculty Enhancement grant.

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Correspondence to Hui Zhang.

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Lin, J., Zhang, H. Accelerating visual communication of mathematical knot deformation. J Vis 23, 913–929 (2020). https://doi.org/10.1007/s12650-020-00663-w

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  • DOI: https://doi.org/10.1007/s12650-020-00663-w

Keywords

  • Knot untanglement
  • View selection
  • Least squares fitting
  • Parallelization