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What is the implicit and/or parametric equations for hyperbolic shapes?

Posted 1 year ago
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I'm working on programming an implicit surface plotter, and one of my test surfaces is the Astroidal Ellipsoid (or the Hyperbolic Octahedron). I was able to successfully get this to work, however I wanted to extend this to further hyperbolic shapes, such as the Hyperbolic Icosahedron and Hyperbolic Dodecahedron.

The issue is that I cannot seem to find equations for the hyperbolic icosahedron or dodecahdron, despite there being a simple implicit and parametric equations for the hyperbolic octahedron. I would like something in the form of "0 = f(x,y,z)" or "x=F(u,v), y=G(u,v), z=H(u,v)". It doesn't have to be in Cartesian coordinates (so polar or cylindrical coordinates will be fine), as long as I have a simple equation I can copy and paste into my program.

Where can I find the equations I'm looking for? Thank you for your help!

9 Replies

I've never heard of this before, but this might be a good next step to explore. I will get into modifying the parametric equations and see if I get anywhere, and see how that works out. I remember a couple months ago I accidentally came up with something like a 3D deltoid (hyperbolic tetrahedron), I'll see if I can replicate that and post it if I do.

A mathematician told me that the defining feature of hyperbolic polyhedra is that the curvature of the surface is -1. don't know if that helps figure out an equation.

It's possible there are no equations, since this reference

uses Newton's method to construct the hyperbolic polyhedra

Yes I saw that while I was looking for the equations. It could be that I'm not using the software correctly, but all I see is this (image shown at the bottom of this reply). After examining the source code, it seems like all it does is plot hardcoded lines and does not use a specific equation.

I also discovered this from my searching, but unfortunately this is also a dead end. The code simply loads a platonic solid and applies a hyperbolic map on it (enter image description here).

From the above demo, I realized that the equations I have for the Hyperbolic Octahedron do not exactly mimic what is happening in the demo. For instance, the implicit equation is:

0 = x^(2/3) + y^(2/3) + z^(2/3) + r^(2/3)

Changing the exponents between 0 and 1 would "hyperbolize" the octahedron as shown in the above demo, but as the exponent tends to infinity the surface approximates a square, unlike the above demo where bulbs are formed.

enter image description here

Igor Rivin was able to plot them in Mathematica

so his research may help

It is the dodecahedron and icosahedron that are being sought.

Thanks for the reply!

I have not, but I'm unsure of how this will solve my problem. I don't have any equations to begin with, what variables am I eliminating?

Have you tried eliminating the parametric variables?

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