Transformation of \(\mathbb{R} \times \mathbb{R}\) in \(\mathbb{R} \times \mathbb{R}\) in which the Cartesian representation corresponds to a rotation in a geometric plane.
Formulas
- The rule of a rotation \(r_O\) of 90° centered on the origin point \(O\) of the Cartesian plane, in the positive direction (counter-clockwise), is \(r_O : (x, y) ↦ (−y, x)\).
The rule of a rotation \(r_O\) of 180° centered on the origin point \(O\) of the Cartesian plane, in the positive direction (counter-clockwise) is \(r_O : (x, y) ↦ (−x, −y)\).
The rule of a rotation \(r_O\) of 270° centered on the origin point \(O\) of the Cartesian plane in the positive direction (counter-clockwise), is \(r_O : (x, y) ↦ (y, −x)\). - For a rotation \(r_O\) of 90° centered on the origin point \(O\) of the Cartesian plane, the transformation matrix is \(\begin{bmatrix}0 & −1\\1 & 0\end{bmatrix}\), so that the coordinates \((x’, y’)\) of a point \(P(x, y)\) by this rotation will be given by \(\begin{bmatrix}0 & −1\\1 & 0\end{bmatrix}\times \begin{bmatrix}y \\x\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}\).
- For a rotation \(r_O\) of 180° centered on the origin point \(O\) of the Cartesian plane, the transformation matrix is \(\begin{bmatrix}−1 & 0\\0 & −1\end{bmatrix}\), so that the coordinates \((x’, y’)\) of a point \(P(x, y)\) by this rotation will be given by \(\begin{bmatrix}−1 & 0\\0 & −1\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}\).
- For a rotation \(r\) of \(theta°\) centered on the origin point in a Cartesian plane, the transformation matrix is \(\begin{bmatrix}\cos{\theta} & −\sin{\theta}\\\sin{\theta} & \cos{\theta}\end{bmatrix}\), so that the coordinates \((x’, y’)\) of a point \(P(x, y)\) by this rotation will be given by \(\begin{bmatrix}\cos{\theta} & −\sin{\theta}\\\sin{\theta} & \cos{\theta}\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}\).
Example
Here is the Cartesian representation of a rotation of 90° centered on the origin point:
The definition of this rotation can be written as: \(r_O : (x, y) ↦ (−y, x)\) or, in matrix terms: \(\begin{bmatrix}0 & −1\\1 & 0\end{bmatrix}\times \begin{bmatrix}y \\x\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}\).
For example, for a rotation of 90° of the point (−3,6) around the origin point: \(\begin{bmatrix}-1×6\\1×-3\end{bmatrix}=\begin{bmatrix}-6\\-3\end{bmatrix}\)