The study of the differential geometry of surfaces in
3-space has a long and celebrated history. Over the last 20 years a
new approach using techniques from singularity theory has yielded
some interesting results (see, for example
[3, 5, 19] for surveys).
Of course surfaces arise in a number of ways: they are often
defined explicitly as the image of a mapping
f: R2→R3.
Since the subject is differential geometry one
normally asks that these defining mappings are smooth, that is
infinitely differentiable, however it is not true, in any sense,
that most such parametrisations will yield
manifolds. For such mappings have self-intersections, and more
significantly they
may have crosscaps (also known as Whitney umbrellas). Moreover if
we perturb the maps these singularities will persist; that is they are
stable. (See [14, 20] for details.)
Consequently when studying the differential geometry of surfaces
in 3-space there are good reasons for studying surfaces with crosscaps.
In this paper we carry out a classification of mappings from 3-space
to lines, up to
changes of co-ordinates in the source preserving a crosscap. We can
apply our results
to the geometry of generic crosscap points. In [10]
we computed geometric normal
forms for the crosscap and used them to study the dual of the
crosscap. We shall see
that the approach here yields more information than that obtained in
[10], although
the latter has the advantage of being more explicit (in particular
various aspects of the geometry can be compared using the normal forms).
We refer the reader to [16, 18]
for background material concerning singularity theory.