The four types of isometries of R^{2} are (a) translations, (b) reflections, (c) rotations, and (d) glide reflections. A glide reflection is the transformation that results when a reflection is followed by a non-zero translation parallel to the axis of the reflection.
The identity transformation may be regarded as both a translation and a rotation.
A non-zero translation is an orientation-preserving isometry with no fixed point.
A non-identity rotation is an orientation-preserving isometry with a fixed point that is called the center of the rotation. (In this case there is only one fixed point.)
A reflection is an orientation-reversing isometry with a fixed point. (In this case there is a fixed line.)
A glide reflection is an orientation-reversing isometry with no fixed point.
Let f be the isometry of R^{2} that is obtained by following rotation about the origin counter-clockwise through the angle pi/6 with translation by the vector (2, 0).
Explain why f must be a rotation.
Give a geometric construction of the center of rotation for f.
Find the center of rotation for f analytically.
What is the angle of rotation for f ?