Some applications of singularity theory to the geometry of curves and surfaces.

Tari, Farid. (1990) Some applications of singularity theory to the geometry of curves and surfaces. PhD thesis, University of Liverpool.

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Abstract

This thesis consists of two parts. The first part deals with the orthogonal projections of pairs of smooth surfaces and of triples of smooth surfaces onto planes. We take as a model of pairwise smooth surfaces the variety X= {(x, 0, z): x> 0} U {(0, y, z): y> 0} and classify germs of maps R3,3,0 -º R2,2,0 up to origin preserving diffeomorphisms in the source which preserve the variety X and any origin preserving diffeomorphisms in the target. This yields an action of a subgroup xA of the Mather group A on C3 2, the set of map-germs R3,0 -º R2,0. We list the orbits of low codimensions of such an action, and give a detailed description of the geometry of each orbit. We extend these results to triples of surfaces. In the second part of the thesis we analyse the shape of smooth embedded closed curves in the plane. A way of picking out the local reflexional symmetry of a given curve -y is to consider the centres of bitangent circles to the curve. ° The closure of the locus of these centres is called the Symmetry Set of y. We present an equivalent way of tracing the local reflexional symmetry of -r by considering the lines with respect to which a point on y and its tangent line are reflected to another point on the curve and to its tangent line. The locus of all these lines form the dual curve of the symmetry set of -y. We study the singularities occurring on duals of symmetry sets and their generic transitions in 1-parameter families of curves 7. A first attempt to define an analogous theory to study the local rotational symmetry in the plane is given. The Rotational Symmetry Set of a curve y is defined to be the locus of centres of rotations taking a point -y(ti) together with its tangent line and its centre of curvature, to y(t2) together with its tangent line and its centre of curvature. We study the properties of the rotational symmetry set and list the generic transitions of its singularities in 1-parameter families of curves ry. In the final chapter we investigate the local structure of the midpoint locus of generic smooth surfaces

Item Type:	Thesis (PhD)
Depositing User:	Symplectic Admin
Date Deposited:	20 Oct 2023 19:00
Last Modified:	20 Oct 2023 19:07
DOI:	10.17638/03175548
Copyright Statement:	Copyright © and Moral Rights for this thesis and any accompanying data (where applicable) are retained by the author and/or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge.
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3175548