Toratopic coordinates

Discussion of shapes with curves and holes in various dimensions.

Re: Toratopic coordinates

Postby PWrong » Mon Mar 09, 2015 3:11 am

Tigric coordinates are actually the same as the parametric equations for a tiger. There is a focal duocylinder with radii a and b.

x = (a + rho Cos[gamma]) Cos[theta];
y = (a + rho Cos[gamma]) Sin[theta];
z = (b + rho Sin[gamma]) Cos[phi];
w = (b + rho Sin[gamma]) Sin[phi]

The isosurfaces are tricky, I thought I knew them but I had them wrong. Isosurfaces for rho are obviously tigers:
(Sqrt[x^2 + y^2] - a)^2 + (Sqrt[z^2 + w^2] - b)^2 == rho^2


Isosurfaces for gamma are weird cone shapes. We have
(sqrt(x^2 + y^2) - a)/cos(gamma) = rho = (sqrt(z^2 + w^2) - b)/sin(gamma)

From this you can show that it's a surface of the form
tan(gamma) sqrt(x^2 + y^2) + sqrt(z^2 + w^2) = a tan(gamma) - b.
Do we have a name for this? It's clearly related to cones.

Surfaces of constant theta and phi are half-hyperplanes. Unlike the other coordinate systems, they don't start at the axis, they start at 'a' or 'b' units away from the axis respectively. For example, if we fix phi = 0, then x and y can be anything, but z > b and w = 0.
User avatar
PWrong
Pentonian
 
Posts: 1599
Joined: Fri Jan 30, 2004 8:21 am
Location: Perth, Australia

Re: Toratopic coordinates

Postby ICN5D » Wed Mar 18, 2015 1:50 am

Excellent. I've been digesting this over the last week. The torus in toroidal coord is neat to see how it simplifies greatly. Though, I may actually be looking for a general torus in fixed toroidal coordinates. I've seen the general circle in polar, and it's complex. So, I can only imagine how that translates over.

Not to get too carried way with the mantis here, since there's a thread for it, but I'm thinking of how to express the tiger cut of a column of tori, in toroidal coordinates (for starters). The four circles in polar come out rather simple, so hopefully, it's not as bad as it may seem. My idea is to change some of the rotation period symmetry of the tiger cut, to produce three inward-facing tori. It may be better to put this out there, so you know what I'm after. It's also another reason to learn parametric functions, since they apply directly to coordinate systems.

I'd also like to look into those strange cone shapes. Maybe I can explore/animate one? But, I'm not sure how to use the function

(sqrt(x^2 + y^2) - a)/cos(gamma) = rho = (sqrt(z^2 + w^2) - b)/sin(gamma)

How do I set gamma and rho? The x,y,z,w and a,b are straightforward from a tiger.
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1044
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Toratopic coordinates

Postby PWrong » Thu Mar 19, 2015 2:40 am

Since we're looking at an isosurface for gamma, gamma is a constant. So you can simplify that equation to

sqrt(x^2 + y^2) = A sqrt(z^2 + w^2) + B
for some constants A, B. It's probably best to just look at the case A = 1, B = 0, as that will give you the general shape of the thing. It's not the same as the duocylinder, because that has an extra condition that the radius is fixed.
User avatar
PWrong
Pentonian
 
Posts: 1599
Joined: Fri Jan 30, 2004 8:21 am
Location: Perth, Australia

Re: Toratopic coordinates

Postby PWrong » Thu Mar 19, 2015 2:51 am

I just looked at some intersections for A = -1 and B = 3, they look like spinning tops and cushions. For A = 1, B = 0 they just look like cones.
User avatar
PWrong
Pentonian
 
Posts: 1599
Joined: Fri Jan 30, 2004 8:21 am
Location: Perth, Australia

Previous

Return to Toratopes

Who is online

Users browsing this forum: No registered users and 1 guest