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