In my quest to parameterize this, I found:

http://or.nsfc.gov.cn/bitstream/00001903-5/173475/1/1000009340012.pdf

Which has some useful information; however, it's stated that:

a,b > 0 and c,f >= 0 are constants.
it goes on to state that it's a ring cyclide if f < c < a.

How can f be less than c if for a torus c=0 and f=r?
(unless of course f < 0)...

it's a normal torus if a=b=R,  c=0,  and f=r


Currently in the process of expanding the polynomial and grouping to see how a&b
behave when equal, eliminating the c terms, and seeing how f affects the minor
radii of the Dupin cyclide.

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