- 1 Affine transformation of a circle.
- x
^{2}/a^{2}+ y^{2}/b^{2}= 1

defines an ellipse with width 2a and height 2b centered at the origin.

- 2 Kepler’s equation
- r = 1/(1 − e sinθ)

in polar coördinates defines an ellipse with a focus at the origin and eccentricity e.

- 3 String Construction
- Another construction is the set of points where the sum of two distances to the two foci is constant.

- 4 Conic Section
- Then there is the definition the intersection of a cone and a plane.

No two of these definitions are easily proven to be equivalent. The Dandelin spheres are a fairly easy connection between definitions 3 and 4.