Special Features as Identification Clues for Isometries

The isometries of the plane are of five separate types:  the identity, translations, rotations, line reflections or glide reflections.  To make the types separate, we do not include the identity transformation among the translations or rotations and do not include line reflections as glide reflections. Halfturns, or point reflections, are included among the rotations, but they have some special properties.

These types are in turn divided into two categories -- Orientation Preserving (identity, translations, rotations) and Orientation Reversing (line reflections, glide reflections).

Each type of isometry has distinguishing features.  One important feature of an isometry is whether it has any fixed points or invariant lines.  Such features tell a lot about the isometry.  For example, if T is an isometry and A is a fixed point of T (i.e., T(A) = A), then for any other point B, |T(B)A| = |T(B)T(A)| since A is a fixed point, and this = |BA|, since T is an isometry.  In other words T(B) is on the circle through B with center A and triangle ABT(B) is isosceles.

A line k is an invariant line of T if T(k) = k.  This means that the points of k can move to other points of k but that T does not move the line as a whole to a different line. If all the points of k are fixed, then this is called a line of fixed points; it is invariant, also, but more more.

In this next table, this information is organized a different way. The special feature is listed in the first column.  Then in columns two and three are given the orientation-preserving isometries and orientation reversing isometries that have this property.  This organization is useful for deducing the type of an isometry from some geometrical information.