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 Tv, 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 Rc,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 Fc,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 Fc,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 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. 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.
Read more about this topic: Euclidean Plane Isometry
Famous quotes containing the word plane:
“As for the dispute about solitude and society, any comparison is impertinent. It is an idling down on the plane at the base of a mountain, instead of climbing steadily to its top.”
—Henry David Thoreau (18171862)