Critique of dilation property of ellipse

Mathematicians seek to arrange definitions and theorems in an order that is simple yet insightful. The ease with which the equation for a unit circle can be transformed into the equation for an ellipse hints of a simple and insightful approach to the theory of ellipses. If we define an ellipse to be a figure that is a dilation of a circle, then the proof of the rectangular coordinate equation is very easy. On the other hand, proofs of the conic section and cylindrical section properties can be pretty complex.

We still want to explore polar coordinate and parametric equations of ellipses, as well as look for analogies of other properties of circles.