Euclidean Plane Isometry - Classification of Euclidean Plane Isometries

Classification of Euclidean Plane Isometries

It can be shown that there are four types of Euclidean plane isometries. (Note: the notations for the types of isometries listed below are not completely standardised.)

• Translations, denoted by T_v, where v is a vector in R2. This has the effect of shifting the plane in the direction of v. That is, for any point p in the plane,

or in terms of (x, y) coordinates,

• Rotations, denoted by R_c,θ, where c is a point in the plane (the centre of rotation), and θ is the angle of rotation. In terms of coordinates, rotations are most easily expressed by breaking them up into two operations. First, a rotation around the origin is given by

These matrices are the orthogonal matrices (i.e. each is a square matrix G whose transpose is its inverse, i.e. ), with determinant 1 (the other possibility for orthogonal matrices is −1, which gives a mirror image, see below). They form the special orthogonal group SO(2).

A rotation around c can be accomplished by first translating c to the origin, then performing the rotation around the origin, and finally translating the origin back to c. That is,

or in other words,

Alternatively, a rotation around the origin is performed, followed by a translation:

• Reflections, or mirror isometries, denoted by F_c,v, where c is a point in the plane and v is a unit vector in R2. (F is for "flip".) This has the effect of reflecting the point p in the line L that is perpendicular to v and that passes through c. The line L is called the reflection axis or the associated mirror. To find a formula for F_c,v, we first use the dot product to find the component t of p − c in the v direction,

and then we obtain the reflection of p by subtraction,

The combination of rotations about the origin and reflections about a line through the origin is obtained with all orthogonal matrices (i.e. with determinant 1 and −1) forming orthogonal group O(2). In the case of a determinant of −1 we have:

which is a reflection in the x-axis followed by a rotation by an angle θ, or equivalently, a reflection in a line making an angle of θ/2 with the x-axis. Reflection in a parallel line corresponds to adding a vector perpendicular to it.

• Glide reflections, denoted by G_c,v,w, where c is a point in the plane, v is a unit vector in R2, and w is non-null a vector perpendicular to v. This is a combination of a reflection in the line described by c and v, followed by a translation along w. That is,

or in other words,

(It is also true that

that is, we obtain the same result if we do the translation and the reflection in the opposite order.)

Alternatively we multiply by an orthogonal matrix with determinant −1 (corresponding to a reflection in a line through the origin), followed by a translation. This is a glide reflection, except in the special case that the translation is perpendicular to the line of reflection, in which case the combination is itself just a reflection in a parallel line.

The identity isometry, defined by I(p) = p for all points p is a special case of a translation, and also a special case of a rotation. It is the only isometry which belongs to more than one of the types described above.

In all cases we multiply the position vector by an orthogonal matrix and add a vector; if the determinant is 1 we have a rotation, a translation, or the identity, and if it is −1 we have a glide reflection or a reflection.

A "random" isometry, like taking a sheet of paper from a table and randomly laying it back, "almost surely" is a rotation or a glide reflection (they have three degrees of freedom). This applies regardless of the details of the probability distribution, as long as θ and the direction of the added vector are independent and uniformly distributed and the length of the added vector has a continuous distribution. A pure translation and a pure reflection are special cases with only two degrees of freedom, while the identity is even more special, with no degrees of freedom.