November 1997 Presentation Topic (continued)

Solutions to the problems

Suppose that S = { (x,y) : y=f(x)}. Then, as above

Therefore the curve is parametrized by , where

The tangent vector to at is given by , with

Hence

Thus we conclude that is parallel to the tangent vector to S at (x,f(x)) and therefore is tangent to the circle centered at (x,f(x)) with radius t at the point (x,f(x))+t N(x). See figure 7.

Question 2

Suppose S = { y = ax+b }. The unit normal vector is given by

Hence

Let

Then

Question 3

See figure 8.

Question 4

In this case S_t is parametrized by , where

Thus

Hence, if and only if

Only when t > 1/2 does this have solutions, namely .

In particular, this shows that is smooth for t<0, which corresponds to the evolution of the front along -N(x). For t>0, is smooth as long as t < 1/2. The first value of t for which has a singularity is t = 1/2.

Question 5

For , the points on the curve corresponding to these values of x are

These points satisfy the equation

which is curve with a cusp singularity. See figure 9.

The curves are the pictures of the evolution of the front S with time. We can think of this as the intersections of a surface in with the planes defined by the time t. The surface defined in this way has a singularity called a swallowtail, because of its similarity with the tail of the bird. See figure 10.

It is difficult result, which is outside the scope of this lecture, to show that, in two dimensions, the cusp and the swallowtail are the only stable caustics. Stable in the sense that by slightly moving the initial curve S, any other singularity decomposes into these two. In dimensions greater than two there are many other possible stable caustics. Formation of caustics in higher dimensions is, as we can imagine, a very complicated subject and many people have studied it. There are long lists of possible stable caustics. I will make no attempt of discussing this here. For the interested reader I would suggest the references [1] and [2] mentioned below.

I will end this lecture with the remark that a surface with a swallowtail singularity also appears as an algebraic object. Consider the polynomial of fourth degree in

with coefficients depending on Consider the set

I leave as an exercise for the reader to obtain the polynomial equation of this surface: where

The set corresponds to the values of (x,y,z) for which has a double complex root. It also has a swallowtail singularity and looks very much like the one in figure 10. For a discussion of this see [2].

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