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

Abstract : This thesis consits of two parts. The first part deals with the
orthogonal projections of piecewise smooth surfaces and triples of
surfaces onto planes. We take as a model of piecewise smooth
surfaces the variety $X=\{ (x,0,z): x\ge 0\}\cup \{ (0,y,z): y\ge 0\}$ and classify germs of maps $R^3,0\to R^2,0$ 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 Mather subgroup $_X{\cal A}$ on
$C^{\times 2}_3$, the set of map-germs $R^3,0\to R^2,0$. We list
the orbits of low codimensions of such 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 close curves in the plane. A way of piking the local
reflexional symmetry of a given curve $\gamma$ is to consider the
centres of bitangents circles to the curve. The set of all the
centres is a plane curve called the {\it Symmetrey Set} of
$\gamma$. We present an equivalent way of tracing the local
reflexional symmetry of $\gamma$ by considering the lines with
respect to which a point on $\gamma$ and its tangent line are
reflected to another point on the curve and its tangent line. The
locus of all these lines form the dual curve of the symmetry set
of $\gamma$. We study the singularities occurring on the duals of
symmetry sets and their generic transitions in 1-parameter
families of curves $\gamma$.

A first step to define an analogous theory to study the local
rotational symmetry in the plane is given. The {\it Rotational
Symmetry Set} of a curve $\gamma$ is defined to be the locus of
centres taking a point $\gamma(t_1)$ together with its tangent
line and its centre of curvature, to $\gamma(t_2)$ 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
$\gamma$.

In the final chapter we investigate the local structure of the
midpoint locus of generic smooth surfaces.
Mots-clés :
Document type :
Theses
Domain :

https://tel.archives-ouvertes.fr/tel-00001916
Contributor : Farid Tari <>
Submitted on : Monday, November 4, 2002 - 9:53:07 PM
Last modification on : Thursday, March 5, 2020 - 12:06:06 PM
Long-term archiving on: : Friday, April 2, 2010 - 6:12:43 PM

### Identifiers

• HAL Id : tel-00001916, version 1

### Citation

Farid Tari. Some applications of singularity theory to the geometry of curves and surfaces. Mathematics [math]. Liverpool University, 1990. English. ⟨tel-00001916⟩

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