Translations and Glide Reflections

In this section we introduce a new transformation (glide reflection) and review what we learned in earlier labs and assignments about a familiar transformation (translation).

1. Translations

Several ways to define a translation

These are all properties that could be used to define translations.

1. Given two points A and B, the translation by vector AB, T_AB, maps a point P to the point P' such that ABP'P is a parallelogram.
2. Given two points A and B, the translation by vector AB, T_AB, maps a point P to the point P' such that AB and PP' are parallel, the same length, and the same direction.
3. Given two points C and D, the translation by vector 2CD, T_2CD, is the composition H_DH_C, where H_C and H_D are the point reflections (i.e., half-turns) centered at C and D. Another way to describe 2CD is the vector CC', where C' = H_D(C).
4. Given two parallel lines c and d, the translation T_CC'' is the composition R_dR_c, where R_c and R_d denote line reflection in c and d. If C is any point on c, then let C' = R_d(C); then T_CC'(C) = C' and CC' is the vector of the translation.
5. An isometry T is a translation if for any points P and Q, if P' = T(P) and Q' = T(Q), then PP' is parallel to QQ'.
6. Given two points A and B, the translation by vector AB, T_AB, maps a point P to the point P' such that midpoint AP' = midpoint BP.

Comments

• Statement 1 is the usual definition. However, the definition does not make sense if A, B, P are collinear, so a special defintion must be made for that case.
• Statement 2 is intuitive, but it depends on a good mathematical definition of "direction" that does not depend on translations.
• Statement 3 is a theorem that was explored and proved in Lab 8. It is also featured in Brown, section 2.3.
• Statement 4 was proved as part of the earlier exploration of double line reflections.
• Statement 5 is a theorem that will be proved to be equivalent to these other statements, but is not yet proved.
• Statement 6 is a little strange, but it is equivalent to Statement 1 when the points A, B, P are not collinear and it still works when they are collinear, so it is actually perhaps the best choice for the official definition in the set.

2. Glide Reflections

Defining a glide reflection

Notice that there is just one concept of translation, but it shows up in many forms. The same is true of glide reflections

1. Given two distinct points A and B, the glide reflection G_AB is defined thus: for any point P, reflect P in line AB to get P*; then translate P* by vector AB to define G_AB(P) = P'.
2. A glide reflection G_AA' is the composition R_m H_{A, where Rm is reflection in a line m and HA is reflection in a point A not on m. The point A' = Rm(A)}
3. A glide reflection is the composition R_cR_bR_{a, where a, b, c are lines that are the (extended) sides of a triangle.}
4. A glide reflection is the composition of a line reflection R_m with a rotation with center A, provided A is not on the line m.
5. A glide reflection is an isometry with no fixed points and one invariant line.
6. A glide reflection is the composition of a line reflection R_m with a translation whose translation vector is not perpendicular to m.

Unlike the case of translations, these statements are mostly unfamiliar, not proved to be equivalent and somewhat mysterious at this point. The point of this section of the lab is to get an initial familiarity with glide reflections.

We will take Statement 1 as the definition of a glide reflection, specifically:

Definition: A glide reflection of the plane is an isometry of the plane that is a composition TR, where R is reflection in a line m and T is translation by a vector v parallel to m.

Propostion: If G is a glide reflection defined as TR above, then G^2 = the translation T^2.

Proof. Since the vector v is parallel to m, then TR = RT. This is true since for any P not on m, if we set P' = R(P) and P'' = T(P') and also P* = T(P), it can be shown that R(P*) = P'' and thus RT(P) = TR(P). The reason for this is that PP'P''P* is a rectangle. By definition of T, vector P'P'' = vector PP*, so PP'P''P* is a parallelogram. But also, PP' and P'P'' are perpendicular, since v is parallel to m, so this shows that PP'P''P* is a parallelogram.

Theorem: If G is a glide reflection, there is exactly one line n that is an invariant line for G. That line n is the line m of reflection that is used to define G.

Proof: Suppose that n is an invariant line for G. That means that G(n) = n. But then G(G(n) = G(n) = n also. Since G^2 is a translation by 2v, then n is an invariant line for this translation that thus is parallel to v and to m. But if k is a line distinct from m, but parallel to m, then G(k) and k are lines parallel to m that are on opposite sides of m and so are not the same. Thus the onlyi invariant line is m.

Terminology: If G is a glide reflection, G^2 is translation by a vector 2v. The vector is called the glide vector of G. Also, there is one special line, which is the invariant line of G. This we can say that any glide reflection is the composition of reflection in its invariant line followed (or preceded) by translation by its invariant line: G = TR = RT. This may seem redundant, but there are other ways to define glide reflections, but we can always find an invariant line and glide vector.