Full Version: Scherk's second minimal surface script

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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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Unbelieeeeeeeeeeeevable !!

How did you manage to out this formula together ... hats off
and thank you very much, Brian !!

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Thanks a lot Brian!

Very handy formula!!

Mathematics, this almost unknown "savage beast"...for me! (sigh)




Bye!

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Thank's Tom and Marco.
The equations, as shown in Wikipedia, https://en.wikipedia.org/wiki/Scherk_surface, are the same as in the Scherk2 script.

The equations are the comprehensible result of some mathematician managing to integrate the nearly
incomprehensible Weierstrass–Enneper parameterization.
https://en.wikipedia.org/wiki/Weierstrass%E2%80%93Enneper_parameterization

For some other minimal surfaces, like Costa and Gyroid (I think), the integration must be done numerically, programmed with some
program like Mathematica. The programs create a triangle mesh, not NURBS.

- Brian

This Half version looks better, and uses zero. It can be Networked.

Scherk2WWalfToRotate01
_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;3.141592653589793;24;48;1;;;UV

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"nearly incomprehensible Weierstrass–Enneper parameterization"

From my perspective, what you have posted already looks pretty sophitiscated ;-)

Downloaded the last version- Thank you very much !!

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