Parabolic cyclide

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 From:  bemfarmer
8576.1 
Parabolic cyclide:
_FxGraph3D (a/2)*(2*v*v + b*(1-u*u-v*v))/(1+u*u+v*v);(a*u)*(v*v+b)/(1+u*u+v*v);(a*v)*(1+u*u-b)/(1+u*u+v*v);-10;10;-10;10;50;50;1;0.5;1;UV

Network works at this density.
Open circles show up in v mode, or u mode.
The u and v in the parametric equation range over +/- infinity, so the circles are not closed.
The open circles could be replaced with 3 point circles, and lofted...


Parabola1:
_FxGraph3D u*u;0;sqrt(2)*u;-3.536;3.536;0;1;20;0;1;0;1;UV

Parabola2:
_FxGraph3D 0.5 - 0.5*u*u;-u;0;-5;5;0;1;20;0;1;0;1;UV

The parametric formulas for the two parabolas were mathematically manipulated to match the two standard formula formats.

- Brian

Formulas from Wikipedia and:
http://www.math-only-math.com/parametric-equations-of-a-parabola.html
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 From:  mkdm
8576.2 In reply to 8576.1 
Thanks a lot bemfarmer!

Wow! Awesome formula!!

Bye.

- Marco (mkdm)
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 From:  Mike (MGG942)
8576.3 In reply to 8576.1 
Brian, loft seems to work OK on u, or v, curves without any changes to the formula you provided.

Further, an extrusion of either lofted surface will produce a solid.

I like the shape - reminds me of a 4 leaved clover.

Mike.
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 From:  bemfarmer
8576.4 
Here is another version, with a and b controlling the holes/handles.

Parabolic cyclide:
_FxGraph3D (a*u*u+b*v*v)/(1+u*u+v*v);(a-(b-a)*v*v)*u/(1+u*u+v*v);(b-(a-b)*u*u)*v/(1+u*u+v*v);-10;10;-10;10;50;50;1;-1;1;UV

The _ArcToCircle script is a slightly tested quick modification of the LineWeb script, which converts a bunch of circular arcs to circles.
It was utilized to create a very nice parabolic cyclide.
Run the above Parabolic cyclide script once, outputting U curves.
Change colors, and run the script again, outputting V curves.
Select the U curves and run _ArcToCircle. Do Loft of the U circles. The U arcs can be (color) deleted.
Select the V curves and run _ArcToCircle. Do Loft of the V circles. Color delete the V arcs.
Hide the circles, except for the outer circle of the U curves.
Use the outer circle of the U curves to trim and delete the overlapping portions of the V lofted surface.
Join the lofted U surface with the remaining lofted V surface. Make the resulting double hole/double handle surface the same color.
In Right view, select the origin, and make a new circle to trim away outer lobes, preserving the center holes/handles.
This is a brief synopsis of the creation of the .3dm. Most or all of the steps are here.
Of course, periodically save the interim work, just in case of a problem.

- Brian


EDITED: 7 Sep by BEMFARMER


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 From:  bemfarmer
8576.5 
Finally, a ball with two holes.
Placed a sphere around one hole of the parabolic cyclide, to remove the handle, and with another sphere, did some trimming, fillet, rotate copy, etc.

- Brian


EDITED: 7 Sep by BEMFARMER


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 From:  bemfarmer
8576.6 In reply to 8576.5 
.3dm of ball with holes uploaded to previous post.
Cyclides and circles and spheres tend to go together...
- Brian

Two edges fillet well. Their fillet faces were copied to fillet surfaces, rotated twice to the other hole entrance, and used
to trim the other two edges which previously did not fillet with the Fillet command.

EDITED: 7 Sep by BEMFARMER

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