Due to my current inability to render a Dupin cyclide (DC) made of its
equivalent Villarceau circles based on my own reasoning, I'm trying to explore
other avenues.

My current state of things experienced frustration due to finding apparently
flawed implementations of computing the center and radius of a circle in 3d
given 3 points.
I currently have a known-good version.

My present experiment is trying to leverage the "easily parameterizable" nature
of DC's and replicate the following diagram:

https://en.wikipedia.org/wiki/Dupin_cyclide#/media/File:Zyklide-maxwell-abcd.svg

The equations in the accompanying text sometimes seem to work, and sometimes
not.
So do I question my code, or the equations?

And for context:
The equations are presumably written by a mathematician who was for many years a
member of Working Group 2 for Geometry & Algebra of the Department of
Mathematics of the TU Darmstadt, and contributed to MANY articles on Wikipedia
from 2013-2022.

There are issues with signs, circles have negative radii, and I'm trying to
document what is correct and what is not --- but it keeps changing depending on
the values I assign to x1, x2, x3, x4.

Perhaps others can try to replicate the diagram and the channel surface (the
envelope of spheres, which I'm representing as circles so i can still see the
rest of the diagram)

https://en.wikipedia.org/wiki/Dupin_cyclide#/media/File:Zyklide-kanalfl-1-2.svg

- BE

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