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Growth, motion and 1-parameter families of symmetry sets

Published online by Cambridge University Press:  14 November 2011

J. W. Bruce  and
P. J. Giblin
J. W. Bruce
Affiliation:
Department of Mathematics, The University, Newcastle upon Tyne NE1 7RU, U.K.
P. J. Giblin
Affiliation:
Department of Pure Mathematics, The University, Liverpool L69 3BX, U.K.
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Synopsis

Associated to every plane curve there is the locus of centres of circles bitangent to that curve, the so-called symmetry set of the curve. We can view this set as the spine of our curve, which can be recovered by taking the envelope of circles of varying radii along this spine. Varying the symmetry set in some isotopy while keeping the radius function fixed may be viewed as crudely modelling motion of the original curve viewed as a biological object. Fixing the symmetry set and varying the radius function can be considered to model growth crudely. In this paper we shall describe the generic changes in the curves which take place in the process of growth and motion, and outline the corresponding results for spheres centred on a space curve. We also use the idea of a stratified Morse function to describe the generic changes which occur in one parameter families of (full) bifurcation sets in the plane. Applying this to the bifurcation set of distance squared functions we find all the transitions of a symmetry set (and evolute) which occur in a generic isotopy of a plane curve.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 104 , Issue 3-4 , 1986 , pp. 179 - 204
Copyright
Copyright © Royal Society of Edinburgh 1986

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