Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T03:19:22.006Z Has data issue: false hasContentIssue false
  • English
  • Français

Global theorems for symmetry sets of smooth curves and polygons in the plane

Published online by Cambridge University Press:  14 November 2011

T. F. Banchoff  and
P. J. Giblin
T. F. Banchoff
Affiliation:
Department of Mathematics, Brown University, Providence, Rhode Island 02912, U.S.A.
P. J. Giblin
Affiliation:
Department of Pure Mathematics, Liverpool University, Liverpool L69 3BX, U.K.
Get access Rights & Permissions [Opens in a new window]

Synopsis

The symmetry set of a plane curve y is the locus of centres of circles which either (i) are tangent to y at at least two distinct points, or (ii) have at least 4-point (A3) contact with y somewhere. The points (ii) also lie at cusps of the evolute of γ and the symmetry set, together with the evolute, can be studied by regarding their union as a full bifurcation set. For information on symmetry sets, see [2, 4, 5, 7, 8].

Here we shall use a theorem of Ozawa [9] to deduce global formulae relating three features of symmetry sets. The transitions which occur on the symmetry set of a generic one-parameter family of plane curves have been found [4], and more insight into the global formulae can be gained by tracing them through the transitions and verifying that they are invariant (see Section 2). We also briefly introduce symmetry sets of plane polygons in Section 3, and show that, contrary to first impressions, the same global formulae hold there.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 106 , Issue 3-4 , 1987 , pp. 221 - 231
Copyright
Copyright © Royal Society of Edinburgh 1987

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.)

References

1Banchoff, T. F., Gaffney, T. and McCrory, C.. Counting tritangent planes of space curves. Topology 24 (1985), 15–23.CrossRefGoogle Scholar
2Banchoff, T. F. and Giblin, P. J.. Bisupport circles for planar polygons (in prep.).Google Scholar
3Bruce, J. W.. Isotopies of generic plane curves. Glasgow Math. J. 24 (1983), 195–206.CrossRefGoogle Scholar
4Bruce, J. W. and Giblin, P. J.. Growth, motion and 1-parameter families of symmetry sets. Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), 179–204.CrossRefGoogle Scholar
5Bruce, J. W., Giblin, P. J. and Gibson, C. G.. Symmetry sets. Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 163–186.CrossRefGoogle Scholar
6Gaffney, T.. Counting double point singularities. (Preprint, Northeastern University, 1986).Google Scholar
7Giblin, P. J.. Local symmetry in the plane: experiment and theory. Proceedings of conference on computers and geometry, University of Illinois at Chicago, 1986 (to appear).Google Scholar
8Giblin, P. J. and Brassett, S. A.. Local symmetry of plane curves. Amer. Math. Monthly 92 (1985), 689–707.CrossRefGoogle Scholar
9Ozawa, T.. The numbers of triple tangencies of smooth space curves. Topology 24 (1985), 1–13.CrossRefGoogle Scholar