# The Geometry of Reflecting Light Rays

When parallel rays of light enter the coffee cup at an oblique angle, they bounce off the circular side of the cup at various angles, and then hit the bottom of the cup and form the pattern shown on the introductory page. Such patterns formed by reflecting light rays bouncing off objects are called caustics.

We can model this situation by letting the side of the coffee cup be the semicircle of radius 2 shown below, and imagining the parallel light rays coming in horizontal along the lines y=t for various values of t, and reflecting outward.

## Group Activity

Simulate this physical process to see how light rays bounce off the semicircle as the value of t varies.
Physics tells us that when the light ray hits the semicircle, we can compute the angle it will bounce off as follows: draw the normal N to the tangent line T at the point of impact, and the angle of incidence a between the incoming ray and N will be the same as the angle of reflection b between the outgoing ray and N. See the following diagram:

## Question 4

• If the origin is at the center of the semicircle, what are the (x,y) coordinates for the point on the semicircle of radius 2 that the light ray at y=t hits?
• What is the slope of the normal N to the tangent T of the semicircle at that point?
• Can you use the above information to deduce a formula for tan(a) in terms of t, where a is the angle of incidence between the light ray at y=t and the normal N? What about similar formulas for sin(a), cos(a)?
• Explain (first to yourself, then on paper) why the slope of the reflected line should be tan(2a).
• By expressing tan(2a) in terms of sin(a), cos(a), can you come up with a formula just in terms of t for the slope of the line reflected from y=t?

## Question 5

You now have a formula in terms of t for the slope of the reflected line and a point on that line (namely, the point of impact). Therefore using the point-slope formula for a line, you should be able to write down an equation in t,x,y that defines the line reflected from the light ray at y=t.

If you would like to compare with our answer (which may not look exactly the same), we got the equation:

y-t = t sqrt(4-t2)/(2-t2) (x - sqrt(4-t2))

Next: Computing the Envelope of a Family of Curves
Previous: Transforming a Parametrization into an Implicit Algebraic Equation
Up: Introduction
Vic Reiner <reiner@math.umn.edu>
Frederick J. Wicklin <fjw@geom.umn.edu>
Last modified: Mon Apr 15 13:55:06 1996