Up: Euclidean Geometry

Isometries As Products of Reflections

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 2u 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.

Note: The animation first maps the square by translation, and then by composed reflections.

Rotation and Reflection

Any given rotation of 2r 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.

The rotation is then a composition of the two reflections over m and n.

Note: The animation first displays the rotation, then an appropriate composition of reflections.

Glide Reflection and Reflection

As glide reflections is a combination of translation and reflection we can see from the above discussion that a glide reflection can be expressed as the composition of three reflections. The first two reflections compose the translation step of the glide reflection map, and the third reflection is the same reflection as the second step of the glide reflection map.

Composite Isometries And Their Equivalent Reflections

This section contains four animations depicting the various equivalent means of expressing composed isometries.

We begin with a rotation and translation.
The composition above can be expressed as a single rotation (about a different point).
The next animation illustrates the composed rotation and translation as a composition of four reflections. The first two reflections recreate the rotation, and the last two the translation.
As we watch the movie above we can observe that two identical reflections occur one after another (the second and third reflections), effectively cancelling each other, so we are left with two reflectionswhose composition is equivalent to the original composition of rotation and translation.

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Up: Euclidean Geometry

Created: Jul 15 1996 --- Last modified: Jul 15 1996