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