Transformation of \(\mathbb{R} \times \mathbb{R}\) in \(\mathbb{R} \times \mathbb{R}\) whose Cartesian representation corresponds to a reflection of the geometric plane.
Formulas
- The rule for a reflection \(r_x\) over the x-axis in a Cartesian plane is \(s_x : (x, y) ↦ (x, −y)\). The rule for a reflection \(r_y\) over the y-axis in a Cartesian plane is \(r_y : (x, y) ↦ (−x, y)\).
- For a reflection \(r_x\) over the x-axis in a Cartesian plane, the transformation matrix is \(\begin{bmatrix}1 & 0\\0 & −1\end{bmatrix}\), such that the coordinates \((x’, y’)\) of a point \(P(x, y)\) under this reflection are 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 reflection \(r_x\) over the y-axis in a Cartesian plane, the transformation matrix is \(\begin{bmatrix}−1 & 0\\0 & 1\end{bmatrix}\), such that the coordinates \((x’, y’)\) of a point \(P(x, y)\) under the reflection are given by \(\begin{bmatrix}-1 & 0\\0 & 1\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}\).
Example
This is a Cartesian representation of a reflection over the x-axis:
The definition of this reflection may be written as: \(r_x : (x, y) ↦ (x, −y)\) or, in matrix form, as: \(\begin{bmatrix}1 & 0\\0 &−1\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}\).
For example, for the reflection of the point \((-5,3)\) : \(\begin{bmatrix}1×−5\\−1 ×3\end{bmatrix}=\begin{bmatrix}−5\\−3\end{bmatrix}\)