Two Parabolic Mirrors Opposite of Each other

Question

Suppose we have two parabolic mirrors opposite of each other (e.g. $x=y^2$ and $x= -y^2+10$). Also suppose the first mirror is smaller than the second mirror. If a light ray enters into the opening determined by the mirrors, what determines how the light is reflected? Is there a particular mathematical law which desribes these dynamics?

Answer 1

The angle of incidence = the angle of reflection. Each of these angles is taken with respect to the normal of the mirror at the point of incidence.

As a working example, say a light ray comes in from the left at $y=3$ and hits the second mirror of your example. The slope of the tangent there is $-6$, so the slope of the normal is $1/6$. The angle of incidence with respect to this normal is $\arctan{(1/6)}$. The angle of reflection is also $\arctan{(1/6)}$ with respect to the normal, or $2 \arctan{(1/6)}$ with respect to the horizontal. The slope of the reflected ray with respect to the horizontal is found by the tangent double-angle formula to be $12/35$ and the equation of the reflected ray is

$$y-3 = \frac{12}{35} (x-1)$$

You may use this result to determine where (if at all) it will hit the first mirror by setting $x=y^2$.

Answer 2

You have to note that the "angle of incidence=angle of reflectance" still holds in general. There are no perfect mirrors, however, so each reflection will result in some absobtion. This is why a configuration using infinite mirrors like this can be used to make a "light trap" (that is the term we use; i don't know what is correct) used for 0% reflected light standard for color meters.

Answer 3

One standard fact about a parabolic mirror - a beam parallel to the axis will be reflected through the focus, and a beam through the focus will be reflected parallel to the axis. So if you arrange for the foci of the two parabolas to coincide you might get some interesting effects, particularly if your light source is a long way off.