Cyclides

Discussion of shapes with curves and holes in various dimensions.

Cyclides

Postby PWrong » Fri Oct 31, 2014 7:52 am

Cyclides are really cool. It's basically like a quadric except that one of the terms is the square of the radius, so it's actually a quartic.

So a cyclide in 3D would be described by P(x, y, z, r^2) = 0, where P is a second order polynomial and r^2 = x^2 + y^2 + z^2. As far as I can see there's no standard definition for a four dimensional cyclide.

Here's one example that's equivalent to a torus:
(x^2 + y^2 + z^2 - 1)^2 - 2 ( x^2 + y^2 - z^2 ) + 1 = 0

There's a subset of cyclides called Dupin cyclides, which seem to look very similar to toruses, but wider in some parts than others.

There's a few articles about these things, but most of them are very old. I think it would be interesting to classify cyclides the same way we classify quadrics. We could extend the definition to 4D and see how they relate to toratopes. Note that since cylides are quartic surfaces, but ((II)(II)) and (((II)I)I) are octic surfaces, those toratopes will not be cyclides.
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Re: Cyclides

Postby PWrong » Sun Nov 02, 2014 2:21 am

I made an animation of one.

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Re: Cyclides

Postby ICN5D » Mon Nov 03, 2014 11:32 pm

Whoa, look at that. So, is that a 4D cyclide passing through 3D? Im having a tough time trying to visualize how its built, if so. In the animation, what parameter is being adjusted?
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Re: Cyclides

Postby PWrong » Tue Nov 04, 2014 1:24 am

So far all cyclides are just 3D. This gif is just a 3D cyclide with a parameter changing. The equation I used is
(x^2 + y^2 + z^2 - r^2)^2 - 2 (x^2 + r^2) (f^2 + a^2) - 2 (y^2 - z^2) (a^2 - f^2) + 8 a f r x + (a^2 - f^2)^2 = 0,
where r is the minor radius, a is the major radius and f is called the focus. In my case I used r = 1, a = 2 and let f vary from -3 to 3.

But shapes satisfying this equation only make up a small subset of cyclides. Generally to describe a cyclide you need two scalars, two 3D vectors and a 3x3 matrix. So that's 17 constants altogether. Symmetry will reduce this somewhat. I also wrote some code that generates a random cyclide, most of them look like weird dented bean shapes. It's rare to find one that looks like a torus.

I'm not sure if cyclides have ever been properly classified.
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