Hostname: page-component-cb9f654ff-d5ftd Total loading time: 0 Render date: 2025-09-04T09:17:03.401Z Has data issue: false hasContentIssue false
  • English
  • Français

Perpendicular bisectors, duality and local symmetry of plane curves

Published online by Cambridge University Press:  14 November 2011

Peter Giblin  and
Farid Tari
Peter Giblin
Affiliation:
Department of Pure Mathematics, The University of Liverpool, Liverpool L69 3BX, U.K.
Farid Tari
Affiliation:
Department of Pure Mathematics, The University of Liverpool, Liverpool L69 3BX, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

For a smooth, simple closed curve α in the plane, the perpendicular bisector map P associates to each pair of distinct points (p, q) on α the perpendicular bisector of the chord joining p and q. To a pair (p, p), the map P associates the normal to α at p. The set of critical values of this map is the union of the dual of the symmetry set of α and the dual of the evolute. (The symmetry set is the locus of the centres of circles bitangent to α.) We study the mapP and use it to give a complete list of the transitions which take place on the dual of the symmetry set and the dual of the evolute, as α varies in a generic one-parameter family of plane curves.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 1 , 1995 , pp. 181 - 194
Copyright
Copyright © Royal Society of Edinburgh 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Arnold, V. I.. Catastrophe Theory, 3rd edn (Berlin: Springer, 1992).CrossRefGoogle Scholar
2Blum, H.. Biological shape and visual science, I. J. Theoret. Biol. 38 (1973), 205–287.CrossRefGoogle ScholarPubMed
3Brady, M.. Criteria for representations of shape. In Human and Machine Vision, eds Beck, and Rosenfeld, (New York: Academic Press, 1983).Google Scholar
4Bruce, J. W.. Geometry of singular sets. Math. Proc. Cambridge Philos. Soc. 106 (1989), 495–509.CrossRefGoogle Scholar
5Bruce, J. W. and Giblin, P. J.. Growth, motion and 1-parameter families of symmetry sets. Proc. Roy.Soc. Edinburgh Sect. A 104 (1986), 163–186.CrossRefGoogle Scholar
6Bruce, J. W. and Giblin, P. J.. Projections of surfaces with boundary. Proc. London Math. Soc. (3) 60 (1990), 392–416.CrossRefGoogle Scholar
7Bruce, J. W. and Giblin, P. J.. Curves and Singularities (Cambridge: Cambridge University Press 1984, 2nd edn, 1992).Google Scholar
8Bruce, J. W., Giblin, P. J. and Gibson, C. G.. Symmetry sets. Proc. R. Soc. Edinburgh Sect. A 101 (1985), 163–186.CrossRefGoogle Scholar
9Giblin, P. J. and Brassett, S. A., Local symmetry of plane curves. Amer. Math. Monthly 92 (1985), 689–707.CrossRefGoogle Scholar
10Giblin, P. J. and Tari, F.. Local reflexional and rotational symmetry in the plane. Springer Lecture Notes in Mathematics 1462 (1991), 154–171.CrossRefGoogle Scholar
11Leyton, M.. Symmetry-curvature duality. Computer Vision, Graphics and Image Processing 38 (1987), 327–341.CrossRefGoogle Scholar
12Lu, Y.-C.. Singularity Theory and an Introduction to Catastrophe Theory (Berlin: Springer, 1976).CrossRefGoogle Scholar
13Siersma, D.. Singularities of functions on boundaries, corners, etc. Quart. J. Math. Oxford (2) 32 (1981), 119–127.CrossRefGoogle Scholar
14Tari, F.. Some Applications of Singularity Theory to the Geometry of Curves and Surfaces (Ph.D. Thesis, University of Liverpool, 1990).Google Scholar