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### Reflective Properties of a Semicircular Mirror

```
Date: 05/16/2000 at 10:44:33
From: Hunter DG
Subject: Reflective properties of a circle

I would like to know what happens to parallel rays that enter a
semicircle at any angle. Does a semicircle have reflective properties
similar to one half of the x created by the intersection of the
equations y = x and y = -x?

/\
/  \
/____\
/|    |\
/ |    | \
^    V
^    V

Notice the ray (with the arrows) entering the two lines. Once it hits
a point on the line y = x it is reflected as a horizontal line until
it hits a point on the line y = -x and is then reflected back parallel
to the incoming vertical line:

/\
/  \
/>>>>\
/^    V\
/ ^    V \
^    V
^    V
^    V

Will a circle have a similar property, like this? (Excuse the crude
drawing)
____
_/>>>>\_
/ ^    V \
|  ^    V  |
^    V
^    V
^    V
```

```
Date: 05/16/2000 at 12:52:50
From: Doctor Rick

Hi, Hunter.

No, the semicircle doesn't have this "retroreflective" property. You
can use basic geometry to understand some things about reflection in a
circle. My figure is a bit less crude than yours is, but still not
very good -- I hope you can understand it:

********
***__  |   ***
** |\  \_|      **
*   |a\a  |\__     *
*    |  \  |   \__   *
*     |   \a|      \__ *
*     |    \|     __a_\*
*     |     O____/  a /|
|     |        / |
|     |       / b|
|     |      /   |
|     |     /    |

I drew a line entering vertically (parallel to the centerline of the
hemisphere.) It is reflected such that the angle between the reflected
ray and the radius of the circle at the point of reflection equals the
angle between the incoming ray and the radius. (Both angles are "a" .)
The same happens at the second reflection, if there is one. It's easy
to show that all the angles marked "a" are equal. The angle marked "b"
is how far the outgoing ray is from being parallel to the incoming
ray. You can show that:

b = 180 degrees - 4a

For what angle a is angle b equal to 0? If you know a little
trigonometry, you can figure out where the outgoing ray crosses the
centerline, as a function of angle a; you will find that the rays do
not converge at a single focal point, as they would if the reflector
were a parabola.

You can do further explorations concerning the behavior of rays that
aren't parallel to the centerline. Another interesting exploration is
the 3-dimensional version of the angle reflector you drew. The corner
of a cube has the retroreflective property; when I was your age,
"corner reflectors" like this were placed on the moon by the Apollo
astronauts for laser ranging experiments.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Practical Geometry
High School Symmetry/Tessellations

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