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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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References

1Arnold, V. I.. Wavefront evolution and equivariant Morse lemma. Comm. Pure Appl. Math. 29 (1976), 557–582.CrossRefGoogle Scholar
2Arnold, V. I.. Catastrophe Theory (Berlin: Springer, 1984).CrossRefGoogle Scholar
3Blum, H.. Biological shape and visual science (Part I). J. Theoret. Biol. 38 (1973), 205–287.CrossRefGoogle Scholar
4Bruce, J. W., On singularities, envelopes and elementary differential geometry. Math. Proc. Cambridge Philos. Soc. 89 (1981), 43–48.CrossRefGoogle Scholar
5Bruce, J. W., Wavefronts and parallels in euclidean space. Math. Proc. Cambridge Philos. Soc. 93 (1983), 323–333.CrossRefGoogle Scholar
6Bruce, J. W., Generic isotopies of plane curves. Glasgow Math. J. 24 (1983), 195–206.CrossRefGoogle Scholar
7Bruce, J. W.. Generic functions on semi-algebraic sets. Quart. J. Math. Oxford Ser. (2), 37 (1986), 137–165.CrossRefGoogle Scholar
8Bruce, J. W. and Giblin, P. J.. Curves and Singularities (Cambridge University Press, 1984).Google Scholar
9Bruce, J. W., Giblin, P. J. and Gibson, C. G.. Symmetry Sets. Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 163–186.CrossRefGoogle Scholar
10Fidal, D. L. and Giblin, P. J.. Generic 1-parameter families of caustics by reflexion in the plane. Math. Proc. Cambridge Philos. Soc. 96 (1984), 425–432.CrossRefGoogle Scholar
11Gibson, C. G.. Singular points of smooth mappings. Research Notes in Mathematics 25 (London: Pitman, 1979).Google Scholar
12Looijenga, E. J. N.. The complement of the bifurcation variety of a simple singularity. Invent. Math. 23 (1974), 775–792.CrossRefGoogle Scholar
13Mather, J. N.. Generic projections. Ann. of Math. 98 (1973), 226–245.CrossRefGoogle Scholar
14Nadenik, Z.. Die Ungleichungen för die Masszahlen der geschlossenen Kanalflächen. Czechoslovak Math. J. 16 (1966), 296–306.Google Scholar
15Nadenik, Z.. Die Ungleichungen für die Masszahlen der Kanalkörper. Czechoslovak Math. J. 17 (1967), 408–419.Google Scholar
16Terao, H.. The bifurcation set and logarithmic vector fields. Math. Ann. 263 (1983), 313–321.CrossRefGoogle Scholar
17Arnold, V. I.. Singularities of systems of rays. Uspeckhi Mat. Nauk 38 (1983), 77–147. (Russian Math. Surveys 38 (1983), 87–176.)Google Scholar