Ellipses as dilated circles

A dilation applied to a circle produces an ellipse: it is the most natural way to “see” the ellipse as a circle deformed along the two axes.

Example

The circle $x^2 + y^2 = 1$ dilated with $\lambda = 3,\ \mu = 2$ becomes $\left(\tfrac{x}{3}\right)^2 + \left(\tfrac{y}{2}\right)^2 = 1,\qquad\text{that is}\qquad \tfrac{x^2}{9} + \tfrac{y^2}{4} = 1.$ Ellipses can therefore be thought of as dilations of circles.

The unit circle $C$ and the ellipse $E:\ \tfrac{x^2}{9}+\tfrac{y^2}{4}=1$ obtained by dilating the radii by factors $3$ (horizontal) and $2$ (vertical).

Links

Topics: Plane transformations
Concepts: Circle · Dilation · Ellipse
Methods: Homothetic transformations
Skills: Analytic geometry · Sketching a graph