Seeing Singularities Text

Seeing Singularities

The theory of singularities is the investigation of situations where the derivative of a function (or functions!) is zero. In A-level mathematics maxima and minima of graphs are examples of singularities, although they will probably not be called such. Further examples are abundant in all branches of pure and applied mathematics, they will arise anywhere Newton's calculus is used.

We will investigate how singularities occur in the theory of optics, computer vision and in gravitational lensing.
 

Caustics

When light is reflected from a mirror onto another surface a curve of bright light is often seen on the surface. This curve is called a caustic (from the idea of burning). Caustics have been investigated for many years, for example Newton and Huygens in the 17th Century, but it is only in recent years (1960s onwards) that large advances have been made in the theory.

These advances have been in the local description of the curve and how stable the features. Stable here means that the feature remains despite small changes in the situation, e.g. move the mirror or the light source very slightly.

An example of a stable singularity on a caustic is the cusp, this can be seen in coffee cups. See http://www.ballandclaw.com/Caustic for an interactive web site on this caustic.

Caustic curves for many different curves are given at the following web address.
http://www-groups.dcs.st-and.ac.uk/~history/Curves/Curves.html

Looking at bent wires

One of the current hot topics in singularity theory research is computer vision. This is a difficult problem and like a lot of difficult problems what we do in pure maths is take an easier if somewhat less realistic situation. Thus, instead of considering the outline of a two dimensional object (a surface) consider a one-dimensional object (a curve).

With pieces of wire representing the curve one can investigate questions of stability and how to transform one curve into another using simple moves. With a little bit of experimentation one can convince oneself that the only stable singularity is the crossing of two branches of the wire. Anything else can be removed by a small movement of the wire or one's viewing position. The crossing of two curves is called a node. We can also create a cusp but unlike the caustic case it is not stable.

The curve on the retina formed by the wire will enclose a number of regions, this number is connected to the number of nodes on the curve.

Gravitational lensing

In GCSE Physics one is taught that light travels in straight lines. In A-level this is changed to light rays can be bent by gravity. This has some interesting applications. One example is gravitational lensing.

Imagine a very distant body that is emitting light, for example a quasar (these are very bright bodies at the edge of the universe). Now imagine that a body of large mass (for example a galaxy, we're thinking big here!) comes between you and the source of light. Then light from the distant body gets bent so that more light is arriving at your eye than there was before, so the light gets brighter. The situation gets much worse than this, the mass can also produce multiple images of the quasar. Because the massive body is probably of non-uniform density the light gets bent in a number of ways. The brightness will vary in a number of places, again forming a caustic. Again we can see cusps. They are stable; small changes in mass or position give the same overall shape of the caustic.

Singularity theory tell us that the number of cusps on a general caustic is even.

Two gravitational lensing web sites:
http://www.aip.de./~jkw/
http://astro.uchicago.edu/outreach/partners/fair/proc/wiegert/

Dr. Kevin Houston
School of Mathematics
University of Leeds
Leeds
LS2 9JT
U.K.
email:khouston@amsta.leeds.ac.uk