Abstract
Exact algorithms for detecting all rotational and involutional symmetries in point sets, polygons and polyhedra are described. The time complexities of the algorithms are shown to be θ (n) for polygons and θ (n logn) for two- and three-dimensional point sets. θ (n logn) time is also required for general polyhedra, but for polyhedra with connected, planar surface graphs θ (n) time can be achieved. All algorithms are optimal in time complexity, within constants.
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Aho AV, Hopcroft JE, Ullman JD (1974) The design and analysis of computer algorithms. Addison-Wesley, Reading
Akl SG (1978) Comments on: G. Manacher. An application of pattern matching to a problem in geometrical complexity. Inf Process Lett 7: 86
Bykat A (1979) On polygon similarity. Inf Process Lett 9:23–25
Davis LS (1977) Understanding shape: II symmetry. IEEE Systems Man Cybernet 7: 204–212
Friedberg SA, Brown CM (1984) Finding axes of skewed symmetry. Proceedings of the IEEE Conference on Pattern Recognition, pp 322–325
Friedberg SA, Brown CM (1984) Finding axes of skewed symmetry. Proceedings of the IEEE Conference on Pattern Recognition, pp 322–325
Harary F (1969) Graph theory. Addison-Wesley, Reading
Hopcroft JE, Wong JK (1974) Linear time algorithm for isomorphism of planar graphs. Proceedings of the 6th Annual ACM Symposion on Theory of Computing, pp 172–184
Johansen R, Jones N, Clausen J (1984) A method for detecting structure in polyhedra. Pattern Recognition 2:217–225
Knuth DE, Morris JH, Pratt VR (1977) Fast pattern matching in strings. SIAM J Computing 6:323–350
Lee DT, Preparata FP (1984) Computational geometry — a survey. IEEE Trans Comput 33:1072–1101
Lockwood EH, Macmillan RH (1978) Geometric symmetry. Cambridge University Press, Cambridge
Manachar GK (1976) An application of pattern matching to a problem in geometrical complexity. Inf Process Lett 5:6–7
Martin GE (1982) Transform geometry: an introduction to symmetry Springer, New York
Parvi SK, Dutta Majumder D (1983) Symmetry analysis by computer. Pattern Recognition 16:63–67
Preparata FP, Hong SJ (1977) Convex hulls of finite sets of points in two and three dimensions. Commun ACM 20:87–93
Tanimoto SL (1981) A method for detecting structure in polygons. Pattern Recognition 13:387–394
Wolter JD, Volz RA, Woo TC (1985) Automatic generation of gripping positions. IEEE Trans Systems Man Cybernet (in press)
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Wolter, J.D., Woo, T.C. & Volz, R.A. Optimal algorithms for symmetry detection in two and three dimensions. The Visual Computer 1, 37–48 (1985). https://doi.org/10.1007/BF01901268
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DOI: https://doi.org/10.1007/BF01901268