The four Euclidean isometries, reflection, translation, rotation and glide reflection, can be expressed as compositions of reflections. Below you will find some text, pictures and animations illustrating the relationship between reflection and the other isometries. Reflection The first case, reflection, is trivial.
Reflection is the product of one reflection, itself.
Translation and Reflection Given a translation parallel to a line l , that moves a point 2 u units, we can recreate the translation using reflections over parallel lines . Select any two lines m and n perpendicular to l and u units apart from each other. The action of the translation is equivalent to the composition of two reflections, one across m and the other across n . Rotation and Reflection Any given rotation of 2 r radians about a point x can be expressed as a composition of reflections over intersecting lines . Again, select any two lines m and n such that both intersect at x creating an angle of r radians.