"Bald Eagle" <cre### [at] netscapenet> wrote:
> "Tor Olav Kristensen" <tor### [at] TOBEREMOVEDgmailcom> wrote:
>
> > > Have you thought about geometrically inverting the patches to become a cylinder
> > > or a cone?  :D
> >
> > I'm not sure what you mean by "inverting the patches".
> >
> > I have not yet made cones with Bezier patches. But I have made
> > cylinders. This document shows an example:
> >
> >
https://github.com/t-o-k/Maxima-bezier/blob/main/cylinder_made_with_4_rational_bezier_surfaces_3d.pdf
>
> 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.


> 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!
>
> > The attached image shows the 3x3x3 = 3x9 = 27 control points for
> > two of the patches in the animation. Note that the patches do
> > not get very close to the 9 cyan "intermediate" control points.
>
> Ah yes - that makes it much clearer.
>
> Though if you're interpolating between control grids composed of u,w
> coordinates, via splines controlled by u, then technically u and w are
> parametric equations in u even not explicitly written that way.   Right? (?)

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

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".

E.g.:
If you let u be a constant, let v be controlled by the clock - and
vary w with some code, then a bicubic Bezier curve that wiggles
around within the 3x3x3 grid can be shown (in an animation).


> Beautiful work as always.
> I always enjoy the ballet of the math and graphics, and admire and appreciate
> the skill and discipline to make it all happen.
>
> I'm glad you had the free time to put this one together.  :)

Thank you very much Bill!

    =)

--
Tor Olav
http://subcube.com
https://github.com/t-o-k

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