Math 487 Lab 8

Part B. Introduction to Glide Reflections

Experiment B1: How to glide reflect a point with Sketchpad (and make a tool)

In a new sketch, draw a line AB.  We will denote line reflection in line AB by R and we will denote by T the translation by vector AB. The glide reflection G_AB will be defined to be TR.

• Select the line and choose Mark Mirror from the Transform menu. 
• Select A, then B and choose Mark Vector AB from the Transform menu.
• Draw a point P. Reflect P by the marked mirror line to get P'. Then P' = R(P).
• Translate P' by the marked vector to get P''.  Then P'' = T(P') = TR(P). So G(P) = P''.

Question: Do you get the same result if you construct P* = RT(P)?

Make a tool. Hide point P' and select, A, B, the line, and points P and P''. Make a tool. Call it G_AB. The inputs to the tool should be points A and B and and "pre-image C" or something like that. So the object C to be transformed can be any object.

The transformation that takes S to S’’ is the glide reflection G_AB

Experiment B2: Invariants and Midpoints

Invariant line

In a new sketch, draw a line AB as before and draw another line m. Use your tool to construct the image line m' = G_AB(m). Move m around.

What does m' look like when m is perpendicular to line AB?

What does m' look like when m is parallel to line AB?

For what lines is m = m'? (There is only one such line. It is called the invariant line of the glide reflection. It is line AB.)

Midpoints

In a new sketch, again draw line AB. Then draw several points P, Q, R and the images P', Q', R' by G_AB. Now construct midpoints of the segments QQ'', PP, RR'.

• What do you notice about the midpoints? Whenever you have a glide reflection, you can find the invariant line by using this midpoint property.

Experiment B3: Orbits and Footprints

One way to help visualize a transformation T is to look at the images of several points, or the image of a shape.

Another way is start with a shape S (or a point P) and then find not only the image T(S), but also the image of the image T(T(S)), denoted T²(S), and then T³(S), and so on. In principle we can do this for all possible n, both positive and negative. In practice, a few steps tell a lot.

This collection of images is called the T-orbit of S, or informally the T-footprints of S.

Iterating G to obtain the orbit of a point and a segment

Draw a line AB and a point P not on line AB. Let G = G_AB be the glide reflection defined above.

• Construct G(G(P)) = G²(P) and also G³(P) and G⁴(P).

Midpoints and the "special" Invariant Line of G

• Construct the midpoint of segment PP'' = segment PG(P) and also the midpoints of the segments G(P)G²(P), G²(P)G3(P), and G³(P)G⁴(P) . 
• You should find that these midpoints are collinear and equally spaced (by vector AB). See whether you can prove this. (There is an explanation in Brown.).
• Check that the points P, G(P) an G²(P) are collinear if and only if P is on line AB. Otherwise you get a "zig-zag".

Orbit of a Shape

Suppose G is a glide reflection. With Sketchpad, it is easy to start with a shape S and then construct images G(S), G²(S), G³(S), G⁴(S), etc. The set of these images is part of the G-orbit of S.

In this figure the shapes are colored differently, according to whether they = Gⁿ(S) for even or odd n.

• What isometries will map an even shape to other even shapes? What isometries will map an odd shape to other odd shapes?

Triple Line Reflection: A preview

Draw a triangle ABC and a shape S. Reflect S across line BC to get S', reflect across CA to get S'' and reflect across AB to get S'''. Hide S' and S'' and let T(S) = S'''. Make a tool and find the T-orbit of S. Can you use this to "guess" what kind of transformation T is?