REFLECTIONS AND MINIATURE GOLF

Reflections and their composites, called isometries, have an important place in the study of high school geometry. For instance, our definition of congruence is based upon this concept. Two figures are congruent if and only if it can be shown that one figure is the result of an isometry performed on the other.

Once we have reflected a pre-image across a reflection line or mirror, we see that the segment connecting a pre-image point and its image is bisected by the mirror line. In fact, the mirror line is the perpendicular bisector of this segment.

We can apply the fact that the mirror line is the perpendicular bisector of a pre-image and it reflected image in order to perform reflections, rotations, translations, and glide-reflections using a mirra, or compass and straight-edge, Geometer's Sketchpad, or an all-purpose perpendicular finder (the corner of an index card). This is fun, in and of itself, but can we apply this knowledge in the world around us?

Actually people use reflections all the time.

Try this simple activity. Stand directly in front of a mirror. The reflection you see of yourself has changed orientation (your right side has switched with your left side) and you appear to be twice as far from your image as you are from the mirror. In other words, your image appears to be the same distance behind the mirror as you are in front of it. Now move to the left of center, but continue looking at the same point on the mirror.

Do you see yourself? No!

What you see is the reflection of an object that is situated at same angle from the point on the mirror that you are focusing on. Move further to the left, and these angles get even steeper. These angles have special names in the study of light. They are called the angle of incidence and the angle of reflection. This fundamental property is used to help explain the paths of light, sound, and radio waves.

And ordinary people use this property all of the time.

For instance, when a person tosses a non-spinning ball against a smooth wall, the ball will bounce off of the wall and travel in a path (or ray) that is the reflection of the path the ball would have taken, had the wall not obstructed it. When banking a shot off of the back-board, a basketball player is unconciously taking this fact into account (as well as other factors such as the spin of the ball, its velocity, mass and so on).

The games of billiards and miniature golf are also designed around this property.

Take miniature golf, for instance. Most holes are designed so that the golfer cannot aim the golf ball from the tee directly into the hole. Instead, the golfer must bounce the ball off of one or more walls in order to circumvent various obstacles in its path.

How can you determine the angle (or angles) you will need to bounce a ball in miniature golf?

Let's start with a simple example: bouncing the ball off of one hole.

In the diagram below, we see a rectangular miniature golf plot. Suppose we decide to bank the ball off of the east wall.

To determine the correct angle and, therefore, location from which to bounce the ball, we must first reflect the hole (marked Z across the north wall. Using the east wall as the mirror, or reflection line, find the reflection image of point Z. Label this point Z'.

This ray (ray AZ') shows the direction the golf ball would travel if the wall were not in its path.

Construct the segment from point A (for start) to point Z'. Locate the point of intersection of this segment and the east wall.

That's it! You found it! The point of intersection is the point to which you want to aim the golf ball. If you hit the ball without spin, and with the proper velocity to this point, it will bounce off of the wall and go into the hole (point Z).

Show the path of the ball by drawing a line segment from the point of intersection to the hole.

Do you see it? The angle of incidence equals the angle of reflection.

That seems easy enough. Now let's try to bounce the golf ball off of more than one wall.

Look at the miniature golf course hole again. This time, we will hit the ball off of the tee (that's point A). Then we will bounce the ball off of the north, then south, and then east wall before it to roll into the hole.

As with a lot of math problems, in order to solve this one, we will work backwards. In other words, first we will reflect the hole off of the east wall, then the south wall, and then the north wall.

Here we see the hole (point Z) reflected across line marked east wall. Mark this point Z'

Next, reflect Z' across the south wall. We will have to extend this wall, in order to reflect across it. Mark this point Z''.

Finally, reflect point Z'' across the north wall. Mark this point Z'''.

Now we can outline the ball's path.

Working forwards again, construct segment AZ'''. Locate the point of intersection on the north wall.

Next, construct the segment between the point of intersection and point Z''. Locate this point of intersection on the south wall.

Now, construct the segment between this point of intersection and point Z'.

Finally, construct the segment from the point of intersection on the east wall to the hole (point Z).

Voila! You're done.

Try some miniature golf course plots of your own.