Hi! I’d like to know if there’s a simple way to build a double torus (2 holes in one surface, eg. genus 2 surface) that can be point edited and then 3D printed?
Won’t this result in a polysurface? I want to create a single surface with history that I can point edit. The actual shape I’m building is based on a double torus, but needs to be significantly transformed.
A figure 8 can be created with one single surface but unlikely without self intersections. For 3D printing this might not be an issue.
A sketch might help to explain your concept…
And why do you need one single surface?
It’s a workflow preference. In my experience, this makes iterations faster and cleaner.
It’s sort of impossible to model a double torus with one surface…
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Mathematically, it’s doable. Ultimately, I am really hoping for a torus function that has a g-torus option where you pick the number of cores (g) in the surface. I was thinking someone might have a script to use with an algorithm. This is a little beyond my usual Rhino builds, but would be very useful.
How, with a Moebius transformation?
In topology, it’s a genus 2 surface that can be described by an equation (if you’re interested, there are plenty of math sites about them). Since Rhino is ultimately based in math, I was hoping to use this for my design purposes.
Hi Elise,
Any NURBS surface has four edges and four corners. A genus 1 torus can be formed from such, requiring all four edges. There are no edges left to form the second opening of a genus 2 surface.
AFAIK the genus 2 Bolza surface (a double torus) is formed by folding an octagonal surface.
So regardless of the mathematics (or because of it?) you can’t make a double torus from a single NURBS surface.
Regards
Jeremy
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I see a lot of interesting things and here’s one thing by @DanielPiker which could be adapted…
subD can be edited very good and easy by controlpoints.
does the shape stay symetric?
an other aproach -if you are familiar with the math would be implicit surfaces? :
-no controlpoints here… controlpoints are explicit 
double-torus-isopod.gh (17.1 KB)
yes…
but what i meant was the option to define a shape by typing an equation. possible with a custom field in isopod
you would than modify the resulting shape by tweeking the numbers you feed the equation…
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The isopod surfaces are interesting, but I suspect tweaking the equation inputs will take me an infinitely long time to get right.
The form will not remain symmetric and is very organic, which is part of why I’d like to be able to control point edit.
I guess I may just have to get over making this as a single surface (especially if the NURBS won’t do it).
The genus of a surface tells us nothing about the actual geometric shape of the object - there are infinite possibilities for every genus, so you need to think about what geometric properties you want your shape to have. It could help others help you if you give a bit more context and description of the final result you are aiming for.
As Martin and Jeremy pointed out, for g>1, it is impossible for it to be a single 4 sided NURBS surface (you could kink the edges to effectively make it into a sort of octagon, and shape that into a genus 2 surface, but the result would not be properly connected at those kinks).
There is a nice construction which can be used to create radially symmetric constant-mean-curvature surfaces of any genus, where you slice a torus and open it up radially until you can fit another one in:
Unfortunately this doesn’t have a simple parametric equation- the initial patch needs to be solved for numerically.
You could make a similar construction without the CMC property quite simply though.
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I know the genus only gives a general idea of a form with two handles. The rough sketch I submitted above is pretty close, but looks more symmetrical than the final version will be. It also has more contours than as drawn. I was hoping for a single-surface way to build this, but I see that it’s not executable.
Nice forms in the link.
Thanks everyone!
i doubt that torturing yourself and others to create a complex figure that is impossible to do with on single NURBS surface is going ot make your workflow any faster..
Rhino based on math does not mean that you can identify an untrimmed Nurbs Surface with two holes.
a single torus is still possible but a double torus would have to create intersecting regions. math applications or grapher like the grapher from MacOS actually use mesh to display the surfaces, which does not have the inherent restrictions NURBS do.
so in that sense creating a single surface NURBS double torus which has to have 4 sides is simply not possible. you can maybe get away with trimmed surfaces. there are many fast ways to create the exact model you like.
a fast method might be: create your planar curves - PlanarSrf - duplicate surface and move - BlendSrf to create your round edges.
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