Toratopic coordinates

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Toratopic coordinates

Postby PWrong » Wed Nov 05, 2014 1:24 pm

I made some progress with toratopic coordinates. They're very difficult, and I still can't figure out tigric coordinates or ditoroidal coordinates, but I did manage to find and torispherical and spheritoroidal coordinates.

This will probably need a lot of editing:
https://dl.dropboxusercontent.com/u/12660336/maths/Toratopes/Toratopic%20coordinates.pdf

I was surprised to find out that one of the isosurfaces of the spheritoroidal coordinates is a 4D cyclide but not a toratope. It's described by the following equation:
(a2 + x2 + y2 + z2 + w2)2 = 4 a2 x2 + 4 a2 y2 + (4 a2 / sin2(ϕ)) w^2
where a and ϕ are constants. Technically there should also be a w ≥ 0 in there as well.

The z = 0 cut of this gives a pair of intersecting spheres. The w = 0 cut is just a circle in the xy plane. The x = 0 and y = 0 cuts give a pair of interesting shapes with two horns each:
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The horn looks nicer using the parametric equations:
Image
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Re: Toratopic coordinates

Postby ICN5D » Wed Nov 05, 2014 6:25 pm

Hey, I recognize the double horn slice! Thats a bitangent Villarceau section of spheritorus, where the 3D plane touches two points on the surface. Further back in the tiger explained thread, I put some rotate and translate gifs that show it come about, with pics of it.

If it helps, ditorus has four Villarceau sections, and tiger has two identical looking ones. Wikipedia has an article on solving the plane equation for V sections, using torus as example. Hmm, there is definitely a relation, since these multi-tangent cuts run along multiple points on the surface.

The math required to derive the plane equation is complicated, so I found them using rotation equations. Some of them in 6D are amazing looking with 12 tangent points.
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Re: Toratopic coordinates

Postby PWrong » Thu Nov 06, 2014 3:33 am

Cool! That could be really helpful, it means we might expect some of the isosurfaces of the ditorus and tiger to be Villarceau sections.

The horn shape is a special case of the two horn cyclide (see figure 8 in that article)

The spheritoroidal coordinates with constant theta actually lead to a completely new 3D coordinate system. I can't find it in the books by Moon & Spencer or Morse & Feshbach. If the Laplace equation is separable in these coordinates, then that alone might be a publishable result, and it doesn't even involve the fourth dimension.

The problem I'm having at the moment with ditoroidal and tigric coordinates is that I don't have a coordinate system or a set of isosurfaces. If I had one I could derive the other, but only with a lot of work. Instead I have to guess one, derive the other and see if I get something that works. It's slow going, and it's hard to even know if I've found the "right" answer.
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Re: Toratopic coordinates

Postby PWrong » Thu Nov 06, 2014 12:12 pm

Laplace is indeed separable in two horn cyclidal coordinates. The solutions involve hypergeometric functions.
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Re: Toratopic coordinates

Postby PWrong » Thu Nov 06, 2014 12:13 pm

Could you tell me the implicit equations for all the Villarceau sections of the ditorus and tiger?
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Re: Toratopic coordinates

Postby ICN5D » Fri Nov 07, 2014 1:47 am

Yes, but they would be approximations of angles, using sin and cos.

For tiger rotate equation:

((IY)(Iy))
(sqrt(x^2 + (y*sin(a))^2) - 2)^2 + (sqrt(z^2 + (y*cos(a))^2) - 2)^2 -0.5^2 = 0

the values of " a " will be 0.608 and 0.963, in which both points of both v-sections are along the y-axis.


For ditorus rotations:

The Five Villarceau Plane Angles for (((II)I)I)

• (((IY)y)I)
(sqrt((sqrt(x^2 + (y*sin(a))^2) - 2)^2 + (y*cos(a))^2) - 1)^2 + z^2 = -0.5^2
A = 0.723 , 1.318


• (((IY)I)y)
(sqrt((sqrt(x^2 + (y*sin(a))^2) - 2)^2 + z^2) - 1)^2 + (y*cos(a))^2 = -0.5^2
A = 1.048 , 1.4034


• (((II)Z)z)
(sqrt((sqrt(x^2 + y^2) - 2)^2 + (z*sin(a))^2) - 1)^2 + (z*cos(a))^2 = -0.5^2
A = 1.047
--- This one is bitangent to not just points, but two whole circles. First occurrence of this type is with (((II)I)I) .


This will probably work more effectively:

http://en.wikipedia.org/wiki/Villarceau_circles

From this, one could extrapolate the principle to tiger and ditorus. Basically, the 3-plane is the hypotenuse of a right triangle, major diameter leg a or b is distance between points, minor diameter c is short leg.
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Re: Toratopic coordinates

Postby Marek14 » Fri Nov 07, 2014 8:00 am

You could probably get exact values for angles by finding the coordinates of singular points and using them as values for equation.
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Re: Toratopic coordinates

Postby PWrong » Fri Nov 07, 2014 2:32 pm

Thanks, these might come in handy. A general theory of Villarceau sections would be really cool, especially if I look at 5 and higher dimensional toratopic coordinates and find that this isn't a coincidence.

I'm starting to think maybe there is no single appropriate coordinate system that could be called ditoroidal or tigric coordinates. There may be infinitely many that fit the requirements. The ultimate test is whether they are Laplace-separable. But it's highly likely that Laplace is only separable in quadric and cyclidal coordinates. That's what the books seem to imply, although in 61 years noone has apparently bothered to prove it.
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Re: Toratopic coordinates

Postby PWrong » Fri Nov 07, 2014 2:34 pm

In any case I've given up on ditoroidal and tigric coordinates until I've done a lot more work on the quartic toratopes. Octic toratopes are just too difficult.
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Re: Toratopic coordinates

Postby PWrong » Sun Nov 09, 2014 4:02 pm

And very soon after saying that, I actually found it! I realised that as tau goes to infinity, the ditorus shrinks and approaches a fixed torus. That information restricted the possible coordinate systems so that I could find a set of ditoroidal coordinates that work perfectly.

However, it doesn't seem to be Laplace-separable. I won't know for sure for a while, there's a huge expression that I have to try to break down.
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Re: Toratopic coordinates

Postby PWrong » Tue Nov 11, 2014 7:31 am

Update:

I still can't figure out tigric coordinates. I suspect that if I can find them, I'll be able to generalise to all toratopic coordinates.

Torispherical coordinates are separable, which is good news. The solution involves an ordinary differential equation that Mathematica can't solve, which is interesting because it actually looks relatively simple.

Spheritoroidal coordinates don't seem to be separable, which is very surprising. I expected it to separate because it's a cyclidal system.

Ditoroidal coordinates probably don't separate, which isn't surprising.
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Re: Toratopic coordinates

Postby PWrong » Wed Nov 12, 2014 7:26 am

I thought of a way to get a better version of torispherical coordinates, that might be separable. It will generalise to all quartic toratopes in a simple way. It might even work for ditoroidal, but I'm not getting my hopes up.
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Re: Toratopic coordinates

Postby ICN5D » Wed Nov 12, 2014 7:41 am

So what does it actually mean to be separable? I've checked out the articles, but it's over my head. Is it the x= , y= , z= , definitions?
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Re: Toratopic coordinates

Postby PWrong » Wed Nov 12, 2014 10:29 am

It's kind of a complicated topic. When I say separable I actually mean R-separable, which is more complicated again. I'm not sure how much calculus you know, but I'll have a go at explaining separable without the R-.

Laplace's equation in 3D is a partial differential equation with infinitely many solutions. The solutions are scalar functions f(x,y,z). The initial conditions restrict things so that only one solution is possible. An example of an initial condition might be "The solution is equal to a constant everywhere on this sphere". The shape of the condition tells you what coordinate system to use, so with this example we'd use spherical coordinates.

Let's say we want to solve Laplace's equation in Cartesian coordinates. We look for solutions of the form
f(x,y,z) = X(x) Y(y) Z(z).
So we have a product of functions of only one variable each. This is called separation of variables. This breaks Laplace's equation into three separate "ordinary differential equations", which you can solve with second or third year calculus techniques (or in practice, with Mathematica).

Now let's say the initial conditions have spherical symmetry, so we convert to spherical coordinates. This transforms Laplace's equation into a totally different equation involving r, θ, φ instead of x, y, z. Now we solve this by supposing the solution is a product of three functions of these variables.
f(r, θ, φ) = R(r) Θ(θ) Φ(φ)
Again, this breaks down the problem into three easier problems.

Sometimes if a coordinate system is too complicated, this method doesn't break down the system sufficiently, and you can't get a solution. We say that the Laplace equation is separable in a coordinate system if separation of variables reduces the equation to three (or n, in our case) ordinary differential equations.
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Re: Toratopic coordinates

Postby PWrong » Wed Nov 12, 2014 10:32 am

The x=, y=, z= thing is the definition of the coordinate system.
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Re: Toratopic coordinates

Postby PWrong » Mon Nov 17, 2014 6:39 am

I've made heaps of progress. I still have to write everything up in Latex but I'll post it here when I'm done.

I've generalised the quartic toratopic coordinates, and found a general solution. The solution involves hypergeometric functions of tau, and Legendre functions of all the angles. It will take a bit of work to figure out all the constants, but that probably won't be necessary. The basic idea is there.

I found the coordinate system for the tiger, it turns out you can just adapt the parametric equations. Unfortunately tigric coordinates are not Laplace-separable, as I expected. However the equation comes out much nicer than ditoroidal coordinates.

It would be good, and probably not too difficult to get a generalised toratopic coordinate system.

What I'd really like to do now is prove the conjecture (which is simply stated as a fact without proof in the book on this subject) that the only Laplace-separable coordinate systems are quadric and cyclidal. That might be very difficult though. If the conjecture is true, then I've already solved Laplace's equation in every torotopic coordinate system possible. If it's false, then there's a huge, fascinating and completely unknown solution space.

It might be possible to find some special potential function such that Schrodinger's equation is separable in tigric coordinates. That would be cool but probably a waste of time unless the potential function is quite simple. It's more likely to be contrived and boring.
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Re: Toratopic coordinates

Postby ICN5D » Mon Nov 17, 2014 6:24 pm

Do you think there may be some application to the curl function, with tigric coordinates? On some other thread, we were talking about 4D electromagnetism, and the hopf fibration. It was the comparison of these two that made me think a little bit:


Flow of Magnetic Field lines
Image


Rotating Tesseract
Image


As you can see in both instances, everything gets squeezed down the middle, and umbrellas out around the outside, in the flowing direction. Could a magnetic field relate Hopf fibration, which can be expressible in tigric coordinates?
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Re: Toratopic coordinates

Postby PWrong » Tue Nov 18, 2014 2:03 am

I think the connection between the magnetic field and the rotating tesseract is pretty tenuous. A projection of a rotating cube appears to do the same thing. But I'm sure there would be applications to 4D magnetism. Certain physicists or engineers living in four dimensions would have to learn all of these coordinate systems and how to separate various PDE's in them. They'd have the solutions in a table, or a book like Moon & Spencer's "Partial Differential Equations", but a lot longer.

I think at one point we figured out Maxwell's equations in 4D. You need some tensor theory to do it right, because the magnetic field is described by a bivector. The electromagnetic tensor is easy to generalise to higher dimensions (see the two matrices in the first answer here)

I don't actually know much about the Hopf fibration. How do you express it in tigric coordinates?
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Re: Toratopic coordinates

Postby ICN5D » Tue Nov 18, 2014 4:20 am

I'm not sure how to express it. But, I do remember reading about the tiger being an artifact of the hopf fibration. Wondered if there might be some other neat application.
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Re: Toratopic coordinates

Postby PWrong » Tue Nov 18, 2014 7:40 am

If you look at the parametric equations for tiger with radii a,b,r and let a = b = 0, then you get a set of parametric equations that describes a hypersphere, but not the usual set. That's probably what that was about.

In fact for a = b = 0 Laplace is separable. I wonder if the solution is different from hyperspherical coordinates.
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Re: Toratopic coordinates

Postby ICN5D » Wed Nov 19, 2014 3:19 am

In fact for a = b = 0 Laplace is separable. I wonder if the solution is different from hyperspherical coordinates.


Well, that's pretty interesting right there. Which can only lead to more investigation ....
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Re: Toratopic coordinates

Postby PWrong » Wed Nov 19, 2014 4:57 pm

Hopf coordinates are an established thing, but I can't find any indication that anyone has solved Laplace in Hopf coordinates. I've just solved Laplace in both Hopf coordinates and hyperspherical coordinates, and they seem like different solutions. But I'm sure they can't actually be different when the initial conditions are the same, and there's a theorem that guarantees exactly one solution for any set of initial conditions. So this could be interesting.
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Re: Toratopic coordinates

Postby ICN5D » Sun Feb 01, 2015 8:11 pm

I just remembered this thread was here. Now I see what you are trying to do. This is very interesting, and I have a burning curiosity to learn this, and extrapolate to higher toratopes. What I'd like to know are how to convert to and from toratopic <-> cartesian, and how to express a toratope in its own coordinate system.
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Re: Toratopic coordinates

Postby PWrong » Wed Mar 04, 2015 6:23 am

Well I did find coordinate systems for each toratope. My student license for Mathematica has expired so I'll have to get it another way so I can find the files where I worked out the coordinate systems.
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Re: Toratopic coordinates

Postby ICN5D » Wed Mar 04, 2015 5:19 pm

Excellent. I'm really interested in digging deeper into it, and exploring possible uses. Plus, it's something new to learn!
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Re: Toratopic coordinates

Postby PWrong » Sat Mar 07, 2015 2:47 am

Here's the toratopic coordinates for a quartic toratope ((m)n). Pretty simple, it's very similar to bipolar coordinates but with two sets of spherical coordinates tacked on.

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Re: Toratopic coordinates

Postby PWrong » Sat Mar 07, 2015 3:06 am

With these coordinates, the cyclide I mentioned before is not an isosurface after all. It still gives a valid 3D coordinate system where Laplace is separable, so maybe I can publish that at some stage.

The isosurfaces are as follows:

Surfaces of constant tau are ((m)n) toratopes, as you'd expect.

Surfaces of constant sigma are ((n)m) toratopes, which is interesting. If n = 1, then it's a sphere.

Surfaces of constant theta_k and phi_k are various infinite cone cylinders. That is, take a cone with a (k-1)-spherical base, and then extrude it with as many dimensions as you need. For theta_1 and phi_2 specifically, the isosurface is just a half-hyperplane.

EDIT: Changed hyperplane to half-hyperplane.
Last edited by PWrong on Mon Mar 09, 2015 1:59 am, edited 1 time in total.
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Re: Toratopic coordinates

Postby ICN5D » Sun Mar 08, 2015 1:42 am

Whoa, interesting how one of the isosurfaces is the toratopic dual, and then there's the strange cone cylinders.

So, from seeing this, would the degree-8 coordinate systems be based on tetra-polar, with four spherical coordinates tacked on? Then, the tetra-polar would unite into bi-toroidal plus two spherical, and finally to tigric or 3-torus systems?

The other idea I had was how to represent a general torus, in toroidal coordinates. And, if there could be a systematic way to derive a general toratope expressed in its own system. Also, the inverse transform would be very useful to know. There seems to be some very elegant principles behind it:

http://en.wikipedia.org/wiki/Toroidal_coordinates#Inverse_transformation

Using the wiki entry, I came up with:

x = (a(sinh(t)/(cosh(t)-cos(o)))*cos(phi))
y = (a(sinh(t)/(cosh(t)-cos(o)))*sin(phi))
z = (a(sin(o)/(cosh(t)-cos(o))))

Inverse to (x,y,z)

phi = (arctan(y/x))
t - theta = (ln((sqrt((x^2+y^2+a)^2 + z^2))/(sqrt((x^2+y^2-a)^2 + z^2))))
o - sigma = (arccos(-((4a^2-(sqrt((x^2+y^2+a)^2 + z^2))^2-(sqrt((x^2+y^2-a)^2 + z^2))^2)/(2(sqrt((x^2+y^2+a)^2 + z^2))*(sqrt((x^2+y^2-a)^2 + z^2))))))
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Re: Toratopic coordinates

Postby PWrong » Mon Mar 09, 2015 1:33 am

The other idea I had was how to represent a general torus, in toroidal coordinates.


Well this is either trivial or difficult depending on what you mean. Given a fixed toroidal coordinate system, it would be pretty difficult to describe a general torus in that system. But given a fixed torus, you can set up a toroidal coordinate system, and then describe the torus in that sytem.

First you'd choose your x,y,z axes appropriately to make it easy. Then from the wiki article the isosurfaces for tau are

(sqrt(x^2 + y^2) - a coth(tau))^2 + z^2 = a^2 / sinh(tau)^2

This the equation for a torus, so we have
R = a coth(tau)
r = a/sinh(tau)

Rearranging this we get
a = r sinh(tau)
R = r cosh(tau)
tau = arccosh(R/r)
a = sqrt(R^2 - r^2)/r

So the coordinate system is toroidal coordinates with a = sqrt(R^2 - r^2)/r. The variables are tau, sigma and phi. The equation for the original torus in these coordinates is simply tau = arccosh(R/r). Much easier than a torus in Cartesian coordinates.
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Re: Toratopic coordinates

Postby PWrong » Mon Mar 09, 2015 1:58 am

So, from seeing this, would the degree-8 coordinate systems be based on tetra-polar, with four spherical coordinates tacked on? Then, the tetra-polar would unite into bi-toroidal plus two spherical, and finally to tigric or 3-torus systems?

I don't think tetra-polar (I think it would be quadrupolar) coordinates is actually a thing. Tripolar coordinates are a thing, but they're very different from bipolar coordinates.

I've worked out that, just like toroidal coordinates have a focal ring of radius a, ditoroidal coordinates have a focal torus of radii a and b. Ditoroidal coordinates are related to the parametric equations for the focal torus, but with extra sinh and cosh terms.

Here's the Mathematica code for the coordinates:
x = (a + b Sinh[\[Tau]] /(Cosh[\[Tau]] - Cos[\[Sigma]]) Cos[\[Phi]]) Cos[\[Theta]];
y = (a + b Sinh[\[Tau]] /(Cosh[\[Tau]] - Cos[\[Sigma]]) Cos[\[Phi]]) Sin[\[Theta]];
z = Sinh[\[Tau]] /(Cosh[\[Tau]] - Cos[\[Sigma]]) b Sin[\[Phi]];
w = (b Sin[\[Sigma]] )/(Cosh[\[Tau]] - Cos[\[Sigma]]);

Isosurfaces for tau are ditorii.

Isosurfaces for sigma are actually spheritorii (211). So it's not actually a dual thing, it seems to be more about breaking things down.

Isosurfaces for phi are cone cylinders
Isosurfaces for theta are half hyperplanes.
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