Dilation 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 dilation in a geometric plane.
Formulas
- The rule of a dilation [latex]h_O[/latex] centered on the origin point [latex]O[/latex] in the Cartesian plane is [latex]h_O : (x, y) ↦ (kx, ky)[/latex].
- For a dilation [latex]h[/latex] with a scale factor [latex]k[/latex] centered on the origin point of the Cartesian plane, the transformation matrix is [latex]\begin{bmatrix}k & 0\\0 & k\end{bmatrix}[/latex], so that the coordinates [latex](x', y')[/latex] of a point [latex]P(x, y)[/latex] of this dilation will be given by [latex]\begin{bmatrix}k & 0\\0 & k\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x'\\y'\end{bmatrix}[/latex].
Example
Here is the Cartesian representation of a dilation [latex]h[/latex] with centre [latex]O[/latex] and a scale factor of [latex]3[/latex]:
The definition of this dilation can be written as: [latex]r_O : (x, y) ↦ (3x, 3y)[/latex] or, in matrix terms: [latex]\begin{bmatrix}3 & 0\\0 & 3\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x'\\y'\end{bmatrix}[/latex].
For example, for a dilation with centre (0, 0) at the point [latex](7,5)[/latex]: [latex]\begin{bmatrix}3×7\\3 ×5\end{bmatrix}=\begin{bmatrix}21\\15\end{bmatrix}[/latex]