I am looking at the focus-focus definitions of the conics, i.e. defining them as the locus of points with the property that some function of the distance from the point to two foci is a constant.
For an ellipse the sum of distances is constant, and for a hyperbola the difference of distances is constant. The locus of points with the product of distances being constant is a Cassini oval, while the locus of points with the ratio of distances being constant is a circle. Is there a corresponding definition for a parabola? What other curves arise from simple functions of the distance to two points?