Halfturns (also called Point Reflections) are rotations, but special ones because they are so simple. We see how they are related to midpoints, how they combine, and how they are produced as double reflections.
This is a quick introduction to a way of implementing a glide reflection transformation with Sketchpad, including forming a tool. (There are also instructions (for later) for how to do this on paper with straightedge and compass.) The glide reflection will be seen as a special triple line reflection. Then a special midpoint property will be explored.
When a halfturn is composed with a line reflection, this will turn out to be the same triple line reflection as in Part B, so it is easy to deduce what the product is.
Experiment 1: In a new sketch, draw a point O and a line m, not through O. Let H = halturn with center O and let R_m = line reflection in m.
Experiment 2. Also, let n = line through O parallel to m and p = line through O perpendicular to m. If the reflections in these lines are denoted as R_n and R_p, show that R_p R_n R_m = H R_m. This is another way to see that this is a glide reflection.
Since rotations can be expressed as double reflections, with the freedom to pick one of the mirrors, we can use a clever trick to find that the product of two rotations is a rotation (except in a special case then it is a translation) and can find the center and angle of the product.
Draw two points A and B in a new sketch. Go to this link to see one example of how to compose two rotations. Try out another example: Construct the center of rotation of A_30 B_60.
There are several ways to visualize and analyze. Among them:
Review what both these figures will look like for each kind of isometry.