geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

two-dimensional In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...

rotations Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

and reflections 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 ''L''₁. Then reflect ''P''′ to its image ''P''′′ on the other side of line ''L₂''. If lines ''L''₁ and ''L''₂ make an angle ''θ'' with one another, then points ''P'' and ''P''′′ will make an angle ''2θ'' around point ''O'', the intersection of ''L''₁ and ''L₂''. I.e., angle ''POP′′'' will measure 2''θ''. 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

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''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

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

s. Then a rotation can be represented as a matrix, :

\operatorname(\theta) = \begin
  \cos \theta & -\sin \theta \\
  \sin \theta &  \cos \theta
\end,

and likewise for a reflection, :

\operatorname(\theta) = \begin
  \cos 2 \theta &  \sin 2 \theta \\
  \sin 2 \theta & -\cos 2 \theta
\end.

With these definitions of coordinate rotation and reflection, the following four identities hold: :

\begin
  \operatorname(\theta) \, \operatorname(\phi) &= \operatorname(\theta + \phi), \\
  \operatorname(\theta) \, \operatorname(\phi) &= \operatorname(2\theta - 2\phi), \\
  \operatorname(\theta) \, \operatorname(\phi) &= \operatorname\left(\phi + \frac\theta\right), \\
  \operatorname(\phi) \, \operatorname(\theta) &= \operatorname\left(\phi - \frac\theta\right).
\end

These equations can be proved through straightforward

matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...

and application of

trigonometric identities In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvi ...

, specifically the sum and difference 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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

. 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 In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ma ...

. These matrices all have a

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...

whose

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...

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 In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...

: ''O''(2). The following table gives examples of rotation and reflection matrix :

See also