Transformation Geometry — Math 331

February 11, 2004

Four Kinds of Isometries of the Cartesian plane

• Recall that by definition an isometry of R^{n} is a distance-preserving affine transformation. Please refer to the discussion accompanying the assignment due February 4 as well as to exercises involving isometries of R^{2} posed in recent assignments.

• Four types of isometries of R^{2} are (a) rotations, (b) translations, (c) reflections, and (d) glide reflections.

• A rotation has a point called its center. Under a rotation every point is moved through a fixed angle around the circle with the given center on which it lies. A rotation is orientationpreserving.

• A translation is an affine transformation of the form x ↦ x+v. A translation is orientationpreserving.

• A reflection has a line called its axis. Under a reflection every point is sent to its mirror image relative to the axis. The line segment from a point to its image under the reflection is perpendicularly bisected by the axis. A reflection is orientation-reversing.

• A glide reflection is the transformation that results when a reflection is followed with the translation by a non-zero vector parallel to the axis of the reflection. A glide reflection is an orientation-reversing isometry with no fixed point.

• The identity transformation may be regarded as both a translation and a rotation, but is sometimes regarded as neither.

Exercises due Friday, February 13

Let f be the isometry of R^{2} that is obtained by following rotation about the origin counterclockwise through the angle π/6 with translation by the vector (2, 0).

1. Find a 2 × 2 matrix U and a vector v in the plane such that f(x) = Ux + v for each x in R^{2}.

2. Find a point c in R^{2} such that f(c) = c.

3. It being claimed without proof for the moment that every isometry of R^{2} falls into one of the four classes enumerated above, explain from that why this f must be a rotation.

4. Give a geometric construction of the center of rotation for f.

5. What is the angle of rotation for f?