There are several potential definitions of an ellipse and it is not trivial to prove that these definitions are equivalent.
- 1 Affine transformation of a circle.
- x2/a2 + y2/b2 = 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.