# How Algebra Sheds Light on Things

Egbert Brieskorn‘s mathematics books contain a lot more flesh than those structured strictly by definitions, lemmas, and theorems. What might be a hindrance to those looking for a quick grasp of a theory is a treasure trove for others who relish an extra of motivation, history, and examples. Currently, I am reading his book Plane Algebraic Curves (authored together with Horst Knörrer) which, already in the introductory chapter, contains many wonderful examples such as linkages, envelopes, and the Hopf fibration. I will come back to some of these topics in later posts. Today’s post contains a quick glimpse of caustics.

Caustics are produced when light is reflected or refracted by curved surfaces. They are visible as lines of high intensity as can be seen in the reflections of light by a wedding ring (see photo above) or a coffee cup. If you have ever incinerated something (say paper) by using a magnifying glass, you will have a perfect basis for understanding the Greek origin of the name caustic (καυστός: burnt).

In geometrical optics, caustics can be understood as the envelope of the light rays reflected by a surface. An envelope of a family of curves (the reflected light rays in our case) is a curve which is tangent in at least one point to each curve in the family. Here is an illustration of this for the caustic of a circle:

Rays reflected by a ring. The incoming rays (white) are reflected (yellow) and produce the familiar shape of the ring caustic seen in the photo above.

When the reflecting curve is algebraic (described as the zero set of a polynomial) this leads to an algebraic curve as the caustic.

Let’s have a quick look at how to find this curve in the example above. We set up our scene in the plane such that the light source is at the point (0,0) and the ring reflecting the light is a circle of radius one centred at (2, 0). The points $(a, b)$ on this circle satisfy the equation $(a-2)^2+b^2-1=0$. To find the equation of the line of reflected light through the point $(a, b)$, we select a second point $(p, q)$ on that line and set up the two-point form of the equation of a line:

$y-b=\frac{q-b}{p-a}(x-a)$

This gives an equation for all points $(x, y)$ on that line. One way to find a point $(p, q)$ on the line representing the reflected light is to choose a point on the light ray through $(a,b)$. Let’s choose the point $(2a, 2b)$. Then we subtract twice the projection of the line from $(a,b)$ to $(2a, 2b)$ onto the normal of the circle at $(a,b)$. The normal is given by $(a-2, b)$. Thus, the second point on the reflected ray can be found as $(p,q)=(2a,2b)-2<(a,b), (a-2,b)>(a-2,b)$.

This is enough information to set up the equations for the caustic. First, the points $(a, b)$ have to be on the circle reflecting the light rays:

(1) $(a-2)^2+b^2-1=0$

Second, we get an equation describing the reflected ray with respect to a point $(a,b)$ by using the two-point form given above:

(2) $f(a, b, x, y) = (y-b)(a-2a(a-2)^2-2b^2(a-2))-(x-a)(b-2a(a-2)b-2b^3)=0$

Third, we need to express the fact that the caustic is tangent to each curve in the family in at least one point. I am omitting some details here and just tell you that the resulting equation is given by:

(3) $f_ag_b-f_bg_a=0$

The indices indicate partial derivatives. This gives us our last polynomial equation:

$((y-b)(1-6a^2+16a-8-2b^2)+2(2a-2)bx+b-6a^2b+8ab-2b^3)2b - (-4yb(a-2)-a+2a(a-2)^2+6b^2(a-2)-(x-a)(1-2a(a-2)-6b^2))2(a-2) = 0$

Now that we have found a system of algebraic equations, we can find the equation of our caustic by eliminating the variables $a$ and $b$ from the system. I have a lot of respect for the old masters who have done this by hand but for this blog post, I will resort to the help of a computer algebra system. Here, I have asked SageMath to do the job for me:

This gives us the sixth degree equation

$3375x^6 + 10017x^4y^2 + 9909x^2y^4 + 3267y^6 - 37800x^5 - 74736x^3y^2 - 36936xy^4 + 173520x^4 + 204192x^2y^2 + 32400y^4 - 417536x^3 - 241920xy^2 + 555264x^2 + 103680y^2 - 387072x + 110592=0$

and finally, we can plot the result together with our original picture:

The caustic (blue) of a system of reflected light rays (yellow).

If this example has whet your appetite for computational algebraic geometry and you are not already initiated, have a look at the wonderful book Ideals, Varieties, and Algorithms.

There is an interesting, if controversial, theory of dark matter predicting it to form ring caustics in the plane of the Milky Way. In the paper Testing the Dark Matter Caustic Theory Against Observations in the Milky Way, the authors test this theory against actual observations.