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 form 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's internal volume.
While the plane curves above have a single constant for the binary
operation d1 and d2 are inputted into, varying this constant 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 hyperbolic
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 infinity.
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?
Addendum: Expanding on the above ideas of cones and superellipses, let's
consider the surface
x^z + y^z = 1
For 1 <= z <= infinity, this forms a shaft bounded by -1 <= x <= 1, -1
<= y <= 1 that starts as a square with edges of root2 rotated 45-degrees
at z =1, rounds to a circle of radius 1 at z = 2, and rounds out further
to a square of edges length 2 at z = infinity, and for 0 <= z <= 1, it
roundes inward in a way I suspect superficially resembles a phillips
head screw bit... not sure for z < 0, though I have my supicions that
negative exponents cause things to go hyperbolic with asymptotic planes
at x=0 and y = 0.