Isometries are distance preserving transformations. Isometries of
the Euclidean plane are mappings
such that for any two points x and y,
There are four families of isometries:
translation, rotation, reflection and glide reflection.
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.
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
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 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 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.