EUCLIDEAN ISOMETRIES

Introduction

An isometry is a bijective map from the plane onto itself that leaves distance and angles invariant.

There exist four kinds of isometries in Euclidean 2-space. The four isometries are translation, reflection, glide reflection, and rotation. It is interesting to note that the isometries form a group under composition. Furthermore, each isometry can be distinguished by its fixed point behavior and whether or not it preserves orientation.

Below you will find brief descriptions of the isometries and illustrative animations. Keep an eye out for the distinguishing characteristics mentioned above.

The Isometries

Rotation

The rotation mapping is also self-explanatory. Given a fixed point and angle all other points are "rotated" about the fixed point by the given angle.

Rotation has one fixed point (the point about which we rotate) and preserves orientation.

Translation The translation map "shifts" points in the plane. For example, the map may shift the points 4 units along the x-axis, -87 units along the y-axis, or along a diagonal ray, say 5 units in the x direction and 2 units in the y direction. Translation has no fixed points and preserves orientation.
	Reflection The reflection map is, for the most part, self-explanatory. Given a line in the plane, the map "reflects" all points across the line. We can think of the line functioning as a "mirror" reflecting points to the opposite side. For example, if we are reflecting over the line x = 3, the point (4, 0) would be "reflected" to (2, 0). Reflection has an infinite number of fixed points (those on the line we are reflecting over) and reverses orientation.
Glide Reflection Glide reflection is a combination of the above two isometries. Given any line in the plane, the mapping translates the points a given number of units in the direction parallel to the line. The points are then reflected over the line. Glide reflection has no fixed points and is orientation reversing.

Up: Euclidean Geometry

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