Why not stay with the four basic operations and replace one of the points by a straight line?!
We have (with a=distance from the focal point and b=distance from the straight line):
a+b=const -> parabola
a-b=const -> parabola
a/b=const -> conic (a/b>1: hyperbola, a/b=1: parabola, a/b <1 ellipse)
a*b=const. -> ???
Why be content with seven when you can have eleven or more ?
As it looks you like playing around with that kind of curves and also have the time doing so, her some additional ideas.
Stay with simple sum and used weighted distances. Something like two "focal" points 2a+3b=const when a and b are the distances from the "focals".
Or use more than one focal point. What about a curve, where the sum of the distances from three points is constant.
Here is a second animation, varying the constant
But beside all of this we shall not forget, that all these gizmos make only sense, if for these new(?) curves interesting properties (a closed parametric representation, arc length, encloses area), be them mathematical or other, are found.
One more thanks, that you not insert animation into your replies - now I am not in Moscow - I am on my dacha - not far from the place where Tchebyshev was born. Here Internet is very expensive.
Do you know how many built-in Mathcad functions has names based of persons.