Store Studio Observatory Library Geometry Center Info Center Store Studio Observatory Library Geometry Center Information
Create your own Science Me home page!
Library Home Page
Find It!
Site Index

Browse By:
Browse by Subject
Browse by Content Type
Isometry Formulas for the Euclidean Plane
Isometries are distance preserving transformations. Isometries of the Euclidean plane are mappings images/equation1.gif 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 images/equation5.gif 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 images/equation7.gif 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 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.

Info Center | Geometry Center | Library | Observatory | Studio | Store | Science Me

Page last updated Fri May 23 12:43:35 CDT 2003
Comments to

Copyright © Geometry Technologies 1999. All right reserved.