# Mirage

The exhibit “Mirage” shows an interesting optical illusion: A crystal lies in front of an observer as if on a presentation plate. But if you reach for it, you simply — reach into the void. How is this possible? In the following text, we will go into a little more detail about the mathematical background behind this illusion.

The exhibit uses two parabolic reflector to create the strange effect. Such mirrors are also used, for example, in reflecting telescopes and in satellite communications.

#### And now … the mathematics of it:

What is a parabolic mirror and why is it so useful? To understand this, we first need to take a closer look at the parabola: The standard parabola, familiar from school lessons, is described by the equation . We now want to examine the properties of this curve in more detail. To do this, trap a beam of light () from above; this can be described by the equation

Here is a real parameter. In the point this ray is now reflected. At this point the tangent has the normalized direction vector . Thus the normal vector is equal to . Thus, we obtain for the direction vector of the outgoing ray :

Here is the dot product of the vectors and . Thus the precipitating ray is given by the equation

If we now set the coordinate to zero in this equation, we get . This gives

Thus, all incident rays () from above are reflected such that the corresponding outgoing ray passes through the focus . Moreover, we can still calculate the length of the path :

So this distance has exactly the same length as the distance from the point to the horizontal (the so-called directrix of the given parabola) is.

#### Functionality of the exhibit

The “Mirage” exhibit makes use of the above two properties of the parabola. To do this, one first rotates the standard parabola around the axis and thus obtains the equation

of a so-called paraboloid of rotation . This is also used for reflecting telescopes. So what is the idea behind the exhibit? We will describe this in the following. The principle behind it can also be used to transmit other waves (e.g. sound waves):

We take two parabolic mirrors and . Let us say that the mirror is identical with the standard rotational paraboloid . The mirror now faces the mirror at a certain distance . It is thus described by the equation . If one would continue the two parabolic mirrors and arbitrarily far (i.e. for all ) they would intersect in the circle given by the parametrization

for an angle with . For real purposes, however, we need to continue the two mirrors only up to some radius (i.e., for ).

Let and be the respective foci of and . Then, according to the above consideration,

and

We now assume a ray starting at the focal point . Let us say this is given by the equation

Here the parameter measures the rise of the beam . For this to hit the mirror , we need a relatively small slope of . If the ray is now reflected at , the outgoing ray runs afterwards — according to the above considerations — parallel to the -axis at a distance . This ray reflected at is then again reflected by , so that the outgoing ray then passes (again as above) exactly through the focal point . By the same reasoning, a ray starting in — if its slope is large enough (namely ) — will also arrive in after two refexions.

For applications it is still important to know how long the distance covered by the above ray is on its way from to . Let us assume that the ray is reflected at the points and (the straight line is parallel to the axis). Further let be the distance from (and thus ) to the -axis. Then, as explained above,

So if we calculate the total distance that the ray starting in travels from to , we get:

So this distance is independent of .

Now what are the applications of this construction? For example, sound waves (or other waves) can be transmitted well over a long distance in this way: to do this, place the sound source at the point . The sound waves emitted by this source are now reflected at the parabolic mirrors and in the way we just deduced. Thus they concentrate again in the focal point after a certain time, which the sound needs to cover the distance from over the reflection points and to . But this time is proportional to the distance , which has — as derived above — the constant length (independent of the outgoing beam). Thus also no distortion takes place: The transmitted audio signal arrives at point from all directions at the same time. This mode of operation is exploited, for example, in satellite dishes.

The “Mirage” exhibit also uses this trick — but differently from radio wave transmission. Whereas in the latter the distance between the two mirrors is particularly large, in our exhibit it is particularly small: Let’s set (i.e. exactly to the focal length). This time we put our “source” not in the focal point , but in . Similarly, we let the mirrors touch each other, i.e. . Now, if a beam goes

from and if the slope is large enough, is first reflected at and then at and finally “ends” in . Regardless of the rise and direction, the same distance of is always covered. If one cuts out a small circular hole around in , then an object (in the exhibit: a crystal), which is actually in the origin , suddenly appears to the observer as if it were in the point above. This simple trick is used in the exhibit.

At this point it should be noted that if the crystal is mirrored twice, the orientation is also reversed twice, so that the crystal does not appear mirror-inverted either.

If you want to see a nice explanatory video about the above topics, have a look at the video [1] of the YouTuber Mathologer.