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

Posted 1 month 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

Have you tried eliminating the parametric variables?

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?

It is the dodecahedron and icosahedron that are being sought.

Igor Rivin was able to plot them in Mathematica

so his research may help

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

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

uses Newton's method to construct the hyperbolic polyhedra

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.

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.

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