"Tor Olav Kristensen" <tor### [at] TOBEREMOVEDgmailcom> wrote:
https://en.wikipedia.org/wiki/Dupin_cyclide#Dupin_cyclides_and_geometric_inversions
>
> That's some interesting transforms.
>
> It would be an interesting exercise to experiment with them.

I think you get a lot of bang for your buck.
Check out the "Pappus Chain" thread.

> > So you could start with a cylinder, invert it into a torus, and then re-invert
> > it into a cone.  :D  Invert it again into a Dupin cyclide, and then you could
> > invert a final time back to the original cylinder and have QUITE the cyclic
> > animation!

> I'm not sure if I'm able to follow your line of thinking - sorry.

If you follow the diagrams, you'll see that the inversion sphere is a kind of
geometric mirror. And depending upon how the shape and sphere are oriented to
one another, you can get different results.  It's quite the "fun house" mirror.

So my thought was:

A Dupin cyclide can be obtained by inversion of either a ring torus, a cylinder,
or a cone.
Start with a ring torus and invert it to a Dupin cyclide through the appropriate
inversion sphere.
But now, since a Dupin cyclide can be inverted to a cylinder, invert it through
a different sphere to get the cylinder.
Then invert it back to a different Dupin cyclide with yet another inversion
sphere,
and then invert it to a cone with another sphere.
That can be inverted to a Dupin cyclide again,
which can be inverted to the original ring torus.

The real trick is to calculate the parameters of the shapes and the inversion
sphere to get the desired shapes upon each new inversion.

> You can hold any of u, v and w constant and then vary the remaining
> ones with code or by the clock. The result could be a point, a curve,
> a patch or a kind of "solid".

OK, I think I had plain vanilla Bezier surfaces in my mind when you were working
with rational (weighted) Bezier patches.

I will now return you to your regularly scheduled programming.  :)

- BW

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