Title:

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

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 mapgerms 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 1parameter 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 1parameter families of curves ry.
In the final chapter we investigate the local structure of the midpoint locus of
generic smooth surfaces
