Translation 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 translation of the geometric plane.
Formulas
- The rule of a translation [latex]t[/latex] with a vector [latex](a, b)[/latex] in a Cartesian plane is [latex]t_{a, b} : (x, y) ↦ (x + a, y + b)[/latex].
- For a translation [latex]t[/latex] in the Cartesian plane that is defined by a vector [latex]\overrightarrow{t}(a, b)[/latex], the transformation matrix is [latex]\begin{bmatrix}x + a\\y+b\end{bmatrix}[/latex], such that the coordinates [latex](x', y')[/latex] of a point [latex]P(x, y)[/latex] after the translation will be given by [latex]\begin{bmatrix}x + a\\y + b\end{bmatrix}=\begin{bmatrix}x'\\y'\end{bmatrix}[/latex].
Example
This is the Cartesian representation of a translation t by a vector (5, 1).
The definition of this translation may be written as: [latex]t_{5, 1} : (x, y) ↦ (x + 5, y + 1)[/latex] or, in matrix form: [latex]\begin{bmatrix}x + 5\\y + 1\end{bmatrix}=\begin{bmatrix}x'\\y'\end{bmatrix}[/latex]
For example, for the translation of the point [latex](-3,1)[/latex] : [latex]\begin{bmatrix}-3 + 5\\1 + 1\end{bmatrix}=\begin{bmatrix}2\\2\end{bmatrix}[/latex]