2D rotation

In geometry and physics, a 2D rotation is a rotation in two dimensions, commonly on the Euclidean plane. It leaves unchanged a single point, the centre of rotation, often assumed at the origin of the coordinate system. The amount of rotation is represented by the angle of rotation, expressed in units of measurement such as degrees or radians; a signed angle allows for two different senses of rotation: clockwise or anti-clockwise. It corresponds to the 2D special orthogonal group or SO(2).

Formulation

In two dimensions, to carry out a rotation using a matrix, the point Template:Math to be rotated counterclockwise is written as a column vector, then multiplied by a rotation matrix calculated from the angle Template:Math:

$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$.

The coordinates of the point after rotation are Template:Math, and the formulae for Template:Mvar and Template:Mvar are

$\begin{array}{cc}{x}^{\prime }& =x\cos \theta -y\sin \theta \\ y^{\prime }& =x\sin \theta +y\cos \theta .\end{array}$

The vectors $\left[\begin{array}{c}x\\ y\end{array}\right]$ and $\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]$ have the same magnitude and are separated by an angle Template:Mvar as expected.

Template:Anchor Points on the Template:Math plane can be also presented as complex numbers: the point Template:Math in the plane is represented by the complex number

$z=x+iy$

This can be rotated through an angle Template:Mvar by multiplying it by Template:Math, then expanding the product using Euler's formula as follows:

$\begin{array}{cc}{e}^{i\theta }z& =(\cos \theta +i\sin \theta )(x+iy)\\ & =x\cos \theta +iy\cos \theta +ix\sin \theta -y\sin \theta \\ & =(x\cos \theta -y\sin \theta )+i(x\sin \theta +y\cos \theta )\\ & =x^{\prime }+i{y}^{\prime },\end{array}$

and equating real and imaginary parts gives the same result as a two-dimensional matrix:

$\begin{array}{cc}{x}^{\prime }& =x\cos \theta -y\sin \theta \\ y^{\prime }& =x\sin \theta +y\cos \theta .\end{array}$

Since complex numbers form a commutative ring, vector rotations in two dimensions are commutative, unlike in higher dimensions. They have only one degree of freedom, as such rotations are entirely determined by the angle of rotation.

See also

• 2D computer graphics#Rotation
• 3D rotation
• Circle group
• Circular motion
• Instant centre of rotation
• Phase factor
• Polar coordinate system
• Rotation of axes in two dimensions
• Rotations and reflections in two dimensions