Start with two points in the plane, f1 and f2, which we shall call foci.

For any other point p in the plane, there are two associated distances,, 
the distance from p to f1 and from p to f2. Let's call these d1 and d2.

An ellipse is a set of points for which the sum of d1 and d2 is 
constant.

A hyperbola is a set of points for which the absolute difference of d1 
and d2 is constant.

An Oval of Cassini is a set of points for which the product of d1 and d2 
is constant.

What curve is formed by a set of points for which the ratio of d1 and d2 
is constant? As both ellipses and cassini ovals fit into the class of 
curves informally known as ovals, my suspicion is that this curve will 
have a similar relationship to the hyperbola.

Are there other binary operations for which we can input d1 and d2 and 
set the output constant to form an interesting curve?

Because addition and multiplication are communitive and associative, 
it's easy to expand ellipses and cassini ovals to 3 or more foci... is 
there a way to generalize hyperbolas and their ratio-based cousins to 3 
or more foci?

If you expand the definition of an ellipse to all points in 3-space that 
have a constant sum of d1 and d2, you get a prolate spheroid(the foci of 
the generating ellipse don't move in the revolving), and in an oblate 
speroid, the foci of the generating ellipse trace out a circle... 
persumably similar would happen with cassini ovoids... but what about 
with hyperbolas? Does applying the definition of a hyperbola in 
3-dimensions form a hyperboloid of one sheet, a hyperboloid of two 
sheets, or does it from a different surface altogether?

A plane intersects a ellipsoid in an ellipse... if you trace the foci of 
the ellipses formed as a plane passes through an ellipsoid, what shape 
is formed? What shape is formed by taking all the points that are the 
foci of at least one ellipse section of the ellipsoid? For the special 
case of a sphere, the foci of a single plane trace a diameter of the 
sphere an all traces combined form the ball bound by the sphere, but for 
general ellipsoids, the traces are more complex, and I suspect their 
union might miss at least some of the ellipsoid.

While the plane curves above have a single constant for the binary 
operation d1 and d2 are inputted into, varying this coooonstant forms a 
whole family... one way to illustrate the family is to draw all the 
curves in the plane for which the constant is an integer, but another is 
to go 3D and set the constant equal to the z-coordinate... I'm pretty 
sure this produces an elliptical cone for the ellipse, and we could 
probably call the others cones by analogy, but what would a hypebolic 
cone look like? And since conic sections are based on elliptical cones, 
what do sections of cassini or hyperbolic cones look like?

Another generalization of the ellipse is the superellipse, which takes 
the formula for an ellipse and lets the exponents vary... as long as the 
exponents are equal:

Exponent = 2: Standard ellipse.

Exponent > 2: a bulgier than normal super ellipse, becoming a rectangle 
as the exponent goes to infinty.

Exponent between 1 and 2: a flatter than normal ellipse-like curve.

Exponent = 1: a rhombus.

Exponent between 0 and 1: a concave, four-cusped curve similar to an 
asteroid or deltoid... in fact, the asteroid is the case for an aspect 
ratio of one and exponent of 2/3... as the exponent goes to 0, the 
curves approach a pair of perpendicular segments...

What happens if the exponents are less than 0?

Is there a way of generalizing Cassini Ovals and hyperbolas like for 
superellipses?