Geometry Transformations

A transformation is a one-to-one mapping on a set of points. The most common transformations map the points of the plane onto themselves, in a way which keeps all lengths the same. These transformations are called isometries. Another common sort of transformation which does not preserve lengths are dilatations.

There are four isometries in the plane: translationsrotationsreflections, and glide reflections.


translation slides all the points in the plane the same distance in the same direction. This has no effect on the sense of figures in the plane. There are no invariant points (points that map onto themselves) under a translation.



rotation turns all the points in the plane around one point, which is called the center of rotation. A rotation does not change the sense of figures in the plane. The center of rotation is the only invariant point (point that maps onto itself) under a rotation. A rotation of 180 degrees is called a half turn. A rotation of 90 degrees is called a quarter turn.



reflection flips all the points in the plane over a line, which is called the mirror. A reflection changes the the sense of figures in the plane. All the points in the mirror contains all the invariant points (points that map onto themselves) under a reflection.


Glide reflectionsglide reflections

glide reflection translates the plane and then reflects it across a mirror parallel to the direction of the translations. A glide reflection changes the sense of figures in the plane. There are no invariant points (points that map onto themselves) under a glide reflection.


The sense of a figure

SenseThe sense or handedness of a figure refers to the order of points as one goes around the figure. The three triangles in the figure are all congruent, but triangle ABC has the same sense as triangle DEF, and the opposite sense to triangle GHI. A figure has a visible sense if it looks different under a reflection. For example the letter R, if it is reflected, can not be mapped back onto itself by any sequence of translations or rotations. If a figure does not have a visible sense then it is said to have bilateral symmetry. The letter A for example, is symmetrical around a vertical line through its center. If it reflected by a vertical mirror it looks identical. Even if it reflected by a non-vertical mirror, it can be rotated onto itself.


If a figure looks the same under a transformation then it is said to be symmetrical under that transformation. For example, the letter A is symmetrical under a reflection around a vertical mirror through its center. This sort of symmetry is called bilateral symmetry. The letter N is symmetrical under a half turn around the midpoint of its diagonal stroke.

There are many interesting symmetrical patterns which cover the plane, all based on combinations of translationsrotationsreflections, and glide reflections. Another interesting set of symmetrical patterns are frieze patterns.