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