Coordinate rotations and reflections
Encyclopedia
In geometry
, 2D coordinate rotations and reflection
s are two kinds of Euclidean plane isometries
which are related to one another.
A rotation in the plane can be formed by composing a pair of reflections. First reflect a point P to its image P′ on the other side of line L1. Then reflect P′ to its image P′′ on the other side of line L2. If lines L1 and L2 make an angle θ with one another, then points P and P′′ will make an angle 2θ around point O, the intersection of L1 and L2. I.e. angle POP′′ will measure 2θ.
A pair of rotations about the same point O will be equivalent to another rotation about point O. On the other hand, the composition of a reflection and a rotation, or of a rotation and a reflection (composition is not commutative), will be equivalent to a reflection.
The statements above can be expressed more mathematically. Let a rotation about the origin
O by an angle θ be denoted as Rot(θ). Let a reflection about a line L through the origin which makes an angle θ with the x-axis be denoted as Ref(θ). Let these rotations and reflections operate on all points on the plane, and let these points be represented by position vectors. Then a rotation can be represented as a matrix,
and likewise for a reflection,
With these definitions of coordinate rotation and reflection, the following four equations are true:
These equations can be proved through straightforward matrix multiplication
and application of trigonometric identities.
The set of all reflections in lines through the origin and rotations about the origin, together with the operation of composition of reflections and rotations, forms a group
. The group has an identity: Rot(0). Every rotation Rot(φ) has an inverse Rot(−φ). Every reflection Ref(θ) is its own inverse. Composition has closure and is associative, since matrix multiplication is associative.
Notice that both Ref(θ) and Rot(θ) have been represented with orthogonal matrices
. These matrices all have a determinant
whose absolute value
is unity. Rotation matrices have a determinant of +1, and reflection matrices have a determinant of −1.
The set of all orthogonal two-dimensional matrices together with matrix multiplication form the orthogonal group
: O(2).
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, 2D coordinate rotations and reflection
Reflection (mathematics)
In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
s are two kinds of Euclidean plane isometries
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...
which are related to one another.
A rotation in the plane can be formed by composing a pair of reflections. First reflect a point P to its image P′ on the other side of line L1. Then reflect P′ to its image P′′ on the other side of line L2. If lines L1 and L2 make an angle θ with one another, then points P and P′′ will make an angle 2θ around point O, the intersection of L1 and L2. I.e. angle POP′′ will measure 2θ.
A pair of rotations about the same point O will be equivalent to another rotation about point O. On the other hand, the composition of a reflection and a rotation, or of a rotation and a reflection (composition is not commutative), will be equivalent to a reflection.
The statements above can be expressed more mathematically. Let a rotation about the origin
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
O by an angle θ be denoted as Rot(θ). Let a reflection about a line L through the origin which makes an angle θ with the x-axis be denoted as Ref(θ). Let these rotations and reflections operate on all points on the plane, and let these points be represented by position vectors. Then a rotation can be represented as a matrix,
and likewise for a reflection,
With these definitions of coordinate rotation and reflection, the following four equations are true:
These equations can be proved through straightforward matrix multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
and application of trigonometric identities.
The set of all reflections in lines through the origin and rotations about the origin, together with the operation of composition of reflections and rotations, forms a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
. The group has an identity: Rot(0). Every rotation Rot(φ) has an inverse Rot(−φ). Every reflection Ref(θ) is its own inverse. Composition has closure and is associative, since matrix multiplication is associative.
Notice that both Ref(θ) and Rot(θ) have been represented with orthogonal matrices
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
. These matrices all have a determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
whose absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
is unity. Rotation matrices have a determinant of +1, and reflection matrices have a determinant of −1.
The set of all orthogonal two-dimensional matrices together with matrix multiplication form the orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
: O(2).
See also
- Euclidean plane isometryEuclidean plane isometryIn 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...
- dihedral groupDihedral groupIn mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
- Cartan–Dieudonné theorem
- Rotation groupRotation groupIn mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation ...
– 3 dimensions