There are four families of isometries: translation, rotation, reflection and glide reflection.

Translation

Translation by a vector v = (p, q) is given as a vector equation by:

and in coordinates by:

Translations by no fixed points, and are orientation preserving.

Rotation

Translations is are best expressed by a matrix equation. Rotation around the origin through and angle is given by:

Rotations about a point other than the origin are usually best handled as a composition of three transformations. Rotation around v through an angle can be expressed as:

In coordinates, rotation around the origin is given by

The coordinate formula for a rotation around a point v is left as an exercise in composition, though in practice, it is often easier to just do it it in three computational steps. Rotation has exactly one fixed point, and is orientation preserving.

Reflection

Reflection is also nicely represented in a vector equation. If m is a vector lying in the mirror line:

where P is the projection operator. That is, you leave the part in the m direction alone, and reverse the part lying in the direction of the orthogonal to m.

Since projection can be conveniently represented in coordinates, this simplifies in stages to a coordinate formula. If v = (x, y), and m = (p, q),

Reflections through lines not passing through the origin are usually best treated as a composition of functions. If m is a direction vector for a line passing though w,

Reflections have infinitely many fixed points, namely the entire mirror line, and they are orientation reversing.

Glide Reflections

Glide relfections are most naturally expressed as a composition of a translation along a line, followed by a reflection across this line. As a vector equation:

Coordinate formulas are left to the reader. Glide reflections have no fixed points, and are orientation reversing.