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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 c ...
,
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 and
reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
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 ''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 In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
), will be equivalent to a reflection. The statements above can be expressed more mathematically. Let a rotation about the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
''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 s ...
and application of trigonometric identities, 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. 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 a ...
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), ...
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

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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. ...
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Euclidean symmetries Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. It is the name of: Geometry * Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry ...
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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, ...
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Dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
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Cartan–Dieudonné theorem In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an ''n''-dimensional symmetric bilinear space can be described as the composition of at most ''n'' ...
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Rotation group SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is ...
– 3 dimensions {{DEFAULTSORT:Coordinate Rotations And Reflections Euclidean symmetries