Dilation in a Cartesian Plane

Dilation in a Cartesian Plane

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 $$(7,5)$$: $$\begin{bmatrix}3×7\\3 ×5\end{bmatrix}=\begin{bmatrix}21\\15\end{bmatrix}$$