DanA>On the plane, assume the foci are at (-1,0) and (1,0): The ellipse is the locus of points whose sum of distances to the foci is a constant. The ovals of Cassini is the locus of points whose product of distances to the foci is a constant. (See < http://en.wikipedia.org/wiki/File:Line_of_Cassini.svg >.) Can you guess the evolution of the locus of points whose exponentiation of their distances to the foci is a constant, as that constant increases from 0 to, say, 2 ??? (I certainly didn't!) --Dan You mean distance1^distance2 = constant, right? (You left out the hyperbola from distance1 - distance2 = constant.) Re Cassini: Although it has disappeared completely from my personal mail archive (?!!?), the math-fun archive retains a three week(!) discussion, starting ~ 14 Feb 07, on constructing convincing ovals. Amazingly, of all the methods discussed (piecewise circular, Fourier series, polar fitting, rounding triangles, bending steel, the "7-11 curve", sections of surfaces, ...), no one brought up Cassini's ovals! E.g., from sectioning a torus. Nobody thinks about two eggs at a time? But it would make a nice eXcerise (I resisted!) to find the Cassini parameters that best approximate Moss's Egg, by various criteria. Make that two Moss's Eggs. Man, that green racetrack case sure looks flat. But I checked: it's just ±y = Sqrt[-1 - x^2 + 2 Sqrt[1 + x^2]] ~ 1-x^4/8+O(x^6). --rwg