The composition of three ordinary reflections in the plane is generically a glide reflection. A glide reflection can be described by a glide mirror line together with a translation by a vector in the direction of this line. The glide reflection then can also be thought of as reflection in the glide mirror line followed by this translation. The Geometers Sketchpad file Three Reflections can be found in the Math 356 subfolder of the Sketchpad folder on the startup disk of each Macintosh in the Mathlab, Stimson 206. The sketch allows you to visually explore this relationship between three ordinary reflections and the resultant glide reflection.
What you find when doubleclicking on this file are three manipulable mirror lines and a manipulable original triangle. The original triangle is shown in red. Reflection in the red line labeled Mirror 1 transforms this red triangle to a green one. Then reflection in the green line labeled Mirror 2 takes this green triangle to the blue one. Finally reflection in the blue line labeled Mirror 3 transforms the blue triangle into the yellow one.
Thus the effect of the composite glide reflection is to take the red triangle to the yellow one.
The glide mirror line of this glide reflection is shown in purple on the sketch. And the purple segment ending at the point H''' of the yellow triangle is the translation part of the glide reflection.
To manipulate the diagram, first click on the arrow in the upper left of the sketch. This puts sketchpad in selection and manipulation mode. You can then for example, select a mirror line by clicking near it (... staying away from one of the two marked vertices on the line.) Then dragging with your mouse will translate the line. Or select one of the two marked points on one of these lines. Then dragging with your mouse will rotate the mirror line about the other marked point. Similarly you can move or reshape the original triangle.
As you move one of the mirror lines, you change that reflection, and consequently the resultant glide reflection.
Your assignment is to experiment with this and then write a page or two about the relationship between the three mirror lines and the resultant glide reflection. For example, you might address the question of sufficient conditions for the translation part of the glide reflection to be small. Which way does the translation point? When does the glide mirror line appear to be parallel to one of the others? When is there a clear relationship of the glide mirror line to the reflection mirror lines? Or what happens in other natural configurations you notice ...?
As part of your experimentation, look at the following cases. However you needn't write about all of them. You can be selective. The objective of the exercise is to increase your feeling for the geometry, not to be encyclopedic.
Here are some special configurations you should look at. Think about what may be special about the resultant glide reflections.
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