Euclidean plane isometry


In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations, rotations, reflections, and glide reflections.
The set of Euclidean plane isometries forms a group under composition: the Euclidean group in two dimensions. It is generated by reflections in lines, and every element of the Euclidean group is the composite of at most three distinct reflections.

Informal discussion

Informally, a Euclidean plane isometry is any way of transforming the plane without "deforming" it. For example, suppose that the Euclidean plane is represented by a sheet of transparent plastic sitting on a desk. Examples of isometries include:
These are examples of translations, rotations, and reflections respectively. There is one further type of isometry, called a glide reflection.
However, folding, cutting, or melting the sheet are not considered isometries. Neither are less drastic alterations like bending, stretching, or twisting.

Formal definition

An isometry of the Euclidean plane is a distance-preserving transformation of the plane. That is, it is a map
such that for any points p and q in the plane,
where d is the usual Euclidean distance between p and q.

Classification of Euclidean plane isometries

It can be shown that there are four types of Euclidean plane isometries.

Reflections

Reflections, or mirror isometries, denoted by Fc,v, where c is a point in the plane and v is a unit vector in R2. have 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 Fc,v, we first use the dot product to find the component t of p − c in the v direction,
The combination of rotations about the origin and reflections about a line through the origin is obtained with all orthogonal matrices forming orthogonal group O. 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.

Translations

Translations, denoted by Tv, where v is a vector in R2 have the effect of shifting the plane in the direction of v. That is, for any point p in the plane,
A translation can be seen as a composite of two parallel reflections.

Rotations

Rotations, denoted by Rc,θ, where c is a point in the plane, 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
The set of translations and rotations together form the rigid motions or displacements. This set forms a group under composition, the group of rigid motions, a subgroup of the full group of Euclidean isometries.
A rotation can be seen as a composite of two non-parallel reflections.

Glide reflections

Glide reflections, denoted by Gc,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 are a combination of a reflection in the line described by c and v, followed by a translation along w. That is,
The identity isometry, defined by I = 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. 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.

Isometries as reflection group

Reflections, or mirror isometries, can be combined to produce any isometry. Thus isometries are an example of a reflection group.

Mirror combinations

In the Euclidean plane, we have the following possibilities.
Adding more mirrors does not add more possibilities, because they can always be rearranged to cause cancellation.

Recognition

We can recognize which of these isometries we have according to whether it preserves hands or swaps them, and whether it has at least one fixed point or not, as shown in the following table.

Group structure

Isometries requiring an odd number of mirrors — reflection and glide reflection — always reverse left and right. The even isometries — identity, rotation, and translation — never do; they correspond to rigid motions, and form a normal subgroup of the full Euclidean group of isometries. Neither the full group nor the even subgroup are abelian; for example, reversing the order of composition of two parallel mirrors reverses the direction of the translation they produce.
Since the even subgroup is normal, it is the kernel of a homomorphism to a quotient group, where the quotient is isomorphic to a group consisting of a reflection and the identity. However the full group is not a direct product, but only a semidirect product, of the even subgroup and the quotient group.

Composition

Composition of isometries mixes kinds in assorted ways. We can think of the identity as either two mirrors or none; either way, it has no effect in composition. And two reflections give either a translation or a rotation, or the identity. Reflection composed with either of these could cancel down to a single reflection; otherwise it gives the only available three-mirror isometry, a glide reflection. A pair of translations always reduces to a single translation; so the challenging cases involve rotations. We know a rotation composed with either a rotation or a translation must produce an even isometry. Composition with translation produces another rotation, but composition with rotation can yield either translation or rotation. It is often said that composition of two rotations produces a rotation, and Euler proved a theorem to that effect in 3D; however, this is only true for rotations sharing a fixed point.

Translation, rotation, and orthogonal subgroups

We thus have two new kinds of isometry subgroups: all translations, and rotations sharing a fixed point. Both are subgroups of the even subgroup, within which translations are normal. Because translations are a normal subgroup, we can factor them out leaving the subgroup of isometries with a fixed point, the orthogonal group.

Nested group construction

The subgroup structure suggests another way to compose an arbitrary isometry:
  1. If the isometry is odd, use the mirror; otherwise do not.
  2. If necessary, rotate around the fixed point.
  3. If necessary, translate.
This works because translations are a normal subgroup of the full group of isometries, with quotient the orthogonal group; and rotations about a fixed point are a normal subgroup of the orthogonal group, with quotient a single reflection.

Discrete subgroups

The subgroups discussed so far are not only infinite, they are also continuous. Any subgroup containing at least one non-zero translation must be infinite, but subgroups of the orthogonal group can be finite. For example, the symmetries of a regular pentagon consist of rotations by integer multiples of 72°, along with reflections in the five mirrors which perpendicularly bisect the edges. This is a group, D5, with 10 elements. It has a subgroup, C5, of half the size, omitting the reflections. These two groups are members of two families, Dn and Cn, for any n > 1. Together, these families constitute the rosette groups.
Translations do not fold back on themselves, but we can take integer multiples of any finite translation, or sums of multiples of two such independent translations, as a subgroup. These generate the lattice of a periodic tiling of the plane.
We can also combine these two kinds of discrete groups — the discrete rotations and reflections around a fixed point and the discrete translations — to generate the frieze groups and wallpaper groups. Curiously, only a few of the fixed-point groups are found to be compatible with discrete translations. In fact, lattice compatibility imposes such a severe restriction that, up to isomorphism, we have only 7 distinct frieze groups and 17 distinct wallpaper groups. For example, the pentagon symmetries, D5, are incompatible with a discrete lattice of translations.

Isometries in the complex plane

In terms of complex numbers, the isometries of the plane are either of the form
or of the form
for some complex numbers a and ω with |ω| = 1. This is easy to prove: if a = f and ω = f − f and if one defines
then g is an isometry, g = 0, and g = 1. It is then easy to see that g is either the identity or the conjugation, and the statement being proved follows from this and from the fact that f = a + ωg.
This is obviously related to the previous classification of plane isometries, since:
Note that a rotation about complex point p is obtained by complex arithmetic with
where the last expression shows the mapping equivalent to rotation at 0 and a translation.
Therefore, given direct isometry one can solve
to obtain as the center for an equivalent rotation, provided that, that is, provided the direct isometry is not a pure translation. As stated by Cederberg, "A direct isometry is either a rotation or a translation."