Definition. A half-turn is a rotation of the plane about some point through a straight angle.
Observation. The half-turn about the point c is the map x -----> 2 c - x.
The expression 2 c - x, which is a barycentric combination of c and x, does not depend on the choice of an affine coordinate system. Moreover, the tranformation x -----> 2 c - x makes sense in the case when c and x are points of R^{n} for any dimension n. However, this transformation is orientation-reversing when the dimension n is odd.
Proposition. A half-turn about a given point is the product, in either order, of the reflections in any two perpendicular lines intersecting at the center of the half-turn.
Proposition. A glide reflection is always the result of following a half-turn about a point on its axis with the reflection in a line perpendicular to the axis of the glide reflection not passing through the center of the half turn.
Proposition. The composition, in either order, of a reflection with a half turn about a point not on the axis of the reflection is a glide reflection whose axis is the line through the center of the half turn perpendicular to the axis of the reflection.
Calculations involving glide reflections may be facilitated by taking advantage of the fact that there are two different standard ways to represent a glide reflection: the description of the definition (as the composite of a reflection followed by a translation parallel to the axis of the reflection) and the description above.
The three propositions are statements about synthetic plane geometry. Each may be proved easily by beginning with the choice of an optimally convenient Euclidean coordinate system for the plane.
Review the daily handouts looking for statements that you do not understand, that you find wrong, or that you find lacking adequate justification in the context of the course so far.
Bring specific questions.
Web versions of these handouts are available at the URL
http://math.albany.edu:8000/math/pers/hammond/course/mat331/assgt/ .