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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.
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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.

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

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