Full Version: Scherk's second minimal surface script

Show messages:  1-10  11-30  31-34

Hi Tom,
Off the cuff, perhaps setting up a Frame, which "moves the x,y,z axi?"

What are the coordinates and orientation of your required position?

- Brian

Or use Orient command?

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I ran the old sherk2 script again, and applied some different MoI commands.

Lineweb will form a curve between the endpoints of a bunch of curves originating from a single point, with point count = 2.

Rather than arc, Blend was used to connect the ends, then Join. The long lines were trimmed.

Network then works quickly.

Each blend arc and face was then deleted with a plane.

Move and rotate then extended the Sherk2 surface.

- Brian

Image Attachments:
Sherk2Small.png 

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>>I've just checked and can confirm that the MathMod .obj file can be imported into Moi using the importobj script<<

You are right and no problems there. Where I had problems was uploading the obj onto the MOI discussion forum

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Very good and thank you, Brian.

Hope I can follow your steps and will give it a try !

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Brian I realized I cabt follow the steps you layed out.

First, why have you used Scherk2 and not Scherk1 ?

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The Scherk tower is made with Scherk 2nd surface.

Your model resembles a Scherk tower, but reference is made to the Scherk-Collins surface (which is non-minimal)

so after rereading your post, I see that the "saddle surface ring" Scherk-Collins surface:
"can be formed by twisting and warping a singly-periodic Scherk’s minimal surface"
so I'm doing some more reading, both art and math... :-)

https://pdfs.semanticscholar.org/8ac1/d46735688959ec7f1f58a92dd268d0e1388c.pdf

The Scherk1

- Brian

I did try out the mathmod and sculpture generator programs a bit.

I'm going to try stacking some Scherk1 saddles using c=0.99, u = 25, v = 25, 100 pts/curve, trimming the tops and bottoms,
copying above and below, rotate 90 degrees, blending, then network...

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Hi Brian,

Thank you for your input.

Ok thats clear now, I wasnt sure which of your scripts to use.

The original obj was created with this prog
https://people.eecs.berkeley.edu/~sequin/SCULPTS/scherk.html
Great functionality to interactively twist and bend all kind of Scherk towers

I transformed the file generated there into an obj, imported it into MOI
and built uo the curves manually expecting to find a way to create surfaces
out of them,

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There is source code for SculptGen. I've no idea if it can be modified to make nurbs, or which MoI commands could do
equivalent twisting and bending...

After work I'll try to blend together the Scherk1 script output.

- Brian

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The Scherk1 surface can be copied/moved, by PI units in the x direction and also PI units in the y direction, checkerboard fashion.
PI (+/-), is the length of an unwrapped circle of diameter 1 unit.

I read that the Bonnet transformation, with theta = 90 degrees, turns a Scherk1 into a Scherk2 surface. (?How to script?:-)
(Also applies for catenoid/helicoid, and for D, gyroid, P surfaces...)

- Brian

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>>I've no idea if it can be modified to make nurbs<<
Very interested in hearing more about

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Mike;

>>I got an .obj file from MathMod and then made it solid in MeshMixer.<<>>One possibility would be to use retopology tools in 3D Coat or something like Topogun to get a quad polygon layout and then use Max's sub-d to NURBS script to convert that to NURBS.<<

is it opssible in Meshmixer to go form triangle to quads `liKe Michale suggested ?

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I have constructed a Sherk tower manually but the blends are not really smooth.

Is there a way to expand the Scherkscript so it will stack up saddles to build a tower ?

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This is the base unit, for Scherk2 tower, resulting from previously described lineweb, blends, trims, etc.
Just Stack and rotate appropriately.
The complete minimal surface goes to infinity at the poles, I think, precluding a simple "complete" script.
- Brian

The 7Z file is about 1/3 the size of the .zip file. The 7zip unzipper is a free program.

Attachments:
ScherkModule06.7z

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Oh very helpful Brian, thank you very much !
Are you familiar with 3D FX graph ? Is it possible to
get it working with the fomula of a Scherk tower ?

Image Attachments:
SCHERK.jpg 

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Hi Tom,

Let u = r, the radius, which ranges from 0 through 1.
Let v = theta, the angle, ranging from just over 0, to (just under) 2PI ~= 6.2831253.
I tried the script equations with ln, sin, cos, and atan, which mostly reproduced the surface, with some problems with poles v = 0 and v = 2PI.

Using your implicit equation with sin, sinh(x) and sinh(y),
it becomes tedious to enter the equations into the fxgraph3d lines, so I've run out of time...
Best to write the equations in notepad++, as a .js file, and copy them over...then the parenthesis at least can match.
I set u = r and v = theta, as above, which I am not sure is right, but...
set x = u*sin(v)
set y = u*cos(v)

note that sinh(x) = (exp(x) - exp(-x))/2

z = asin((exp(u*sin(v))- exp((-1)*u*sin(v)))/2 * (exp(u*cos(v)) - exp((-1)*u*cos(v)))/2)
Does not work :-(

- Brian

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Thnak you Brian, as I see it is not easy I will see if I succeed !

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For the Scherk2 surface, the equation of a hyperbola in quadrant 1, with vertex at radius of 1 unit, is x*y = 1.
Points (x,y) which produce a z value are contained within the 4 hyperbolic curves.
(In contrast, the Scherk1 surface has a rectangular x,y source, so it was easier to program.)

I think that FXGraph3D could use a numpoints input(?)

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For a mathematician layman what you say is a bit like a Mandarin dialect ;-)
But I am trying to decipher your explanation !

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https://en.wikipedia.org/wiki/Hyperbolic_function

For a hyperbola, angle theta is not used, instead u is used, and is an area.
(For a circle, angle = 2*Area, so either can be used.)

- Brian

http://www.clayton.k12.mo.us/cms/lib/MO01000419/Centricity/Domain/241/ArticleHandout.pdf

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The implicit equation did not work out very well, due to "poles," but here is the classic Weierstrass parameterization, as per Wikipedia.
The surface which results is the same as for the _Scherk2 long .js, .htm script, but lacks symmetry at two ends,
at least partly due to the lack of numpoints parameter in _FXGraph3d (?). So I prefer the long script, after processing, for the Scherk tower approximation.
The FXGraph3d version comes in two halves, due to the exclusion of zero, and replacement with .0001.
The .8 can be increased in value, if U and V count are increased.
The two halves can be networked and joined, offset, edge swept, and joined to a solid, for a super "pringle". A few Fillet tries did not work.

Scherk2WFirstHalf01
_FxGraph3D log((1+u*u+2*u*cos(v)) / (1+u*u-2*u*cos(v))); log((1+u*u-2*u*sin(v)) / (1+u*u+2*u*sin(v))); 2*atan((2*u*u*sin(2*v)) / (u*u*u*u - 1));0;0.8;0.0001;3.141592653589793;21;21;1;;;UV

Scherk2WSecondHalf01
_FxGraph3D log((1+u*u+2*u*cos(v)) / (1+u*u-2*u*cos(v))); log((1+u*u-2*u*sin(v)) / (1+u*u+2*u*sin(v))); 2*atan((2*u*u*sin(2*v)) / (u*u*u*u - 1));0;0.8;0.0001;-3.141592653589793;21;21;1;;;UV

- Brian

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