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Detection of rotational and involutional symmetries and congruity of polyhedra

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Abstract

We propose a simple and efficient general algorithm for determining both rotational and involutional symmetries of polyhedra. It requiresO(m 2) time and usesO(m) space, wherem is the number of edges of the polyhedron. As this is the lower bound of the symmetry detection problem for the considered output form, our algorithm is optimal. We show that a slight modification of our symmetry detection algorithm can be used to solve the related conguity problem of polyhedra.

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References

  • Alt H, Mehlhorn K, Wagener H, Welzl E (1988) Congruence, similarity, and symmetries of geometric objects. Discrete Comput Geom 3:237–256

    Google Scholar 

  • Ball WWR (1892) Mathematical recreations and problems. Macmillan, London

    Google Scholar 

  • Berge C (1985) Graphs. North-Holland, Amsterdam

    Google Scholar 

  • Bykat A (1979) On polygon similarity. In Process Lett 9:23–25

    Google Scholar 

  • Eades P (1988) Symmetry finding algorithms. In: Toussaint, GT (ed) Computational morphology, North-Holland, pp 41–51

    Google Scholar 

  • Eades P, Ng HC (1987) An algorithm for detecting symmetries in line drawings. Ars Combinatoria 23A:95–104

    Google Scholar 

  • Hopcroft JE, Wong JK (1974) Linear time algorithm for isomorphism of planar graphs. Proceedings of 6th Annual ACM Symposion on Theory of Computing, pp 172–184

  • Jiang XY, Bunke H (1992) A simple and efficient algorithm for determining the symmetries of polyhedra. CVGIP: Graph Models Image Process 54:91–95

    Google Scholar 

  • Jiang XY, Bunke H (1993) Detection and application of polyhedral symmetry: review. Proceedings of the 8th Scandinavian Conference on Image Analysis, Tromsø, Norway, pp 345–352

  • Jiang XY, Bunke H (1995) Exploration of object symmetries in computer vision and robotics. In: Bunke H, Kanade T, Noltemeier H, (eds) Modelling and Planning for Sensor Based Intelligent Robot Systems, World Scientific, pp 257–271

  • Lin JC, Chou SL, Tsai WH (1992) Detection of rotationally symmetric shape orientations by fold-invariant shape-specific points. Patt Recogn 25:473–482

    Google Scholar 

  • Ore O (1962) Theory of graphs. Am Math Soc, Providence, RI, p 41

    Google Scholar 

  • Preparata FR, Shamos MI (1985) Computational geometry. Springer Berlin Heidelberg, New York

    Google Scholar 

  • Sugihara K (1984) Annlogn algorithm for detecting the congruity of polyhedra. J Comput Syst Sci 29:36–47

    Google Scholar 

  • Waltzman R (1989) Geometric problem solving by machine visualization. Technical Report CS-TR-2291, University of Maryland, College Park, Md

  • Weinberg L (1996) A simple and efficient algorithm for determining isomorphism of planar triply connected graphs. IEEE Trans Circuit Theory 13:142–148

    Google Scholar 

  • Wolter JD, Woo TC, Volz RA (1985) Optimal algorithms for symmetry detection in two and three dimensions. Vis Comput 1:37–48

    Google Scholar 

  • Zabrodsky H, Peleg S, Arnir D (1992) A measure of symmetry based on shape similarity. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 703–706

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Correspondence to Xiaoyi Jiang.

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Jiang, X., Yu, K. & Bunke, H. Detection of rotational and involutional symmetries and congruity of polyhedra. The Visual Computer 12, 193–201 (1996). https://doi.org/10.1007/BF01782322

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