Math 331 - February 24, 1999

Isometries of the Plane

• A rotation, other than the identity, may be characterized as an isometry with exactly one fixed point that is called its center.

• A reflection may be characterized as an isometry with exactly one fixed line that is called its axis.

• A rotation, other than the identity or a ``half turn'', stabilizes no lines.

• A reflection stabilizes every line that is perpendicular to its axis.

• Two reflections are called parallel if their axes are parallel.

• The composition of two parallel reflections is a translation.

• The composition of two non-parallel reflections is a rotation about the point where their axes meet.

Exercises due Friday, February 26

1. Let f be the affine transformation of the plane defined by

   f(x) = (

   3/5	4/5
   4/5	-3/5

   ) (

   x_{1}

   x_{2}

   ) + (

   1

   2

   ) .

   1. Explain why f is a glide reflection.

   2. What points x of the plane are ``fixed'' by f, i.e., satisfy f(x) = x ?

   3. What lines L in the plane are ``stabilized'' by f, i.e., satisfy the condition that f(x) is on L if x is on L ?

   4. What line is called the axis of f ?

   5. What is the length of displacement along the axis of f ?

   6. Find a reflection g and a translation h parallel to the axis of g such that f = h \circ g.

2. Recall that two triangles DeltaABC and DeltaA'B'C' are congruent if there is at least one isometry f for which one has f(A) = A', f(B) = B', and f(C) = C'. If the triangles are congruent, then how many isometries f satisfy the stated condition?

3. Let A, B, A', B' be any points in the plane, and assume that A and B are not the same point.

   1. Under what circumstances is there at least one isometry f for which f(A) = A' and f(B) = B' ?

   2. If there is at least one isometry satisfying the conditions of the previous question, then how many such isometries are there? Prove your answer.

4. Let U be the matrix of the reflection in the line through the origin and the point ( cos theta, sin theta), i.e.,

   U = (

   cos 2theta	sin 2theta
   sin 2theta	- cos 2theta

   ) .

   Show that the axis of the reflection is never parallel to the one-dimensional column space of the matrix 1 - U. Can you make a general statement about the angle between these two lines?