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Symmetry sets of piecewise-circular curves

  • P. J. Giblin (a1) and T. F. Banchoff (a2)

Synopsis

Piecewise-circular (PC) curves are made up of circular arcs and segments of straight lines, joined so that the (undirected) tangent line turns continuously. PC curves have arisen in various applications where they are used to approximate smooth curves. In a previous paper, the authors introduced some of their geometrical properties. In this paper they investigate the ‘symmetry sets’ of PC curves and one-parameter families of such curves. The symmetry set has also arisen in applications (this time to shape recognition) and its mathematical properties for smooth curves have been investigated by Bruce, Giblin and Gibson. It turns out that the symmetry sets of general one-parameter families of plane curves are mirrored remarkably faithfully by the symmetry sets arising from the much simpler class of PC curves.

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1Arnold, V. I.. Catastrophe theory (Berlin: Springer, 2nd edn, 1986; 3rd edition, 1992).
2Blum, H.. Biological shape and visual science I. J. Theoret. Biol. 38 (1973), 205287.
3Banchoff, T. F. and Giblin, P. J.. Global theorems for symmetry sets of smooth curves and polygons in the plane. Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), 221231.
4Banchoff, T. F. and Giblin, P. J.. On the geometry of piecewise circular curves. Amer. Math. Monthly (to appear).
5Bruce, J. W.. Seeing, the mathematical viewpoint. Math. Intelligencer 6 (1984), 1825.
6Bruce, J. W. and Giblin, P. J.. Growth, motion, and 1-parameter families of symmetry sets. Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), 179204.
7Bruce, J. W., Giblin, P. J. and Gibson, C. G.. Symmetry sets. Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 163186.
8Giblin, P. J. and Brassett, S. A.. Local symmetry of plane curves. Amer. Math. Monthly 92 (1985), 689707.
9Giblin, P. J. and Tari, F.. Local reflexional and rotational symmetry in the plane. Lecture Notes in Mathematics 1462 (1991), 154171.
10Giblin, P. J. and Tari, F.. Perpendicular bisectors, symmetry sets and duals, Preprint, University of Liverpool, 1993.
11Martin, R. R. and Nutbourne, A. W.. Differential Geometry Applied to Curve and Surface Design, Vol. 1 (Chichester: Ellis Horwood, 1988).
12Marciniak, K. and Putz, B.. Approximation of spirals by piecewise circular curves of fewest circular arc segments. Comput. Aided Design 16 (1984), 8790.

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Symmetry sets of piecewise-circular curves

  • P. J. Giblin (a1) and T. F. Banchoff (a2)

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