Conversion from Cartesian to toroidal coordinates

phi = inverse tangent of (y/x). Let q = sqrt(x**2 + y**2). Then the bipolar coordinates can be obtained from the two foci in the xz-plane F1 (-a/2,0) and
F2 (+a/2,0) and the point P (q,z). You have the vertices of the triangle F1 P F2, so you can calculate the distances d1 and d2 and hence the coordinate tau;
the coordinate sigma can then be obtained from the law of cosines. Or have I misinterpreted the definition of the toroidal coordinates?

 

Oh, q is simply the stuff multiplying sin(phi):
q = a sinh(tau)/(cosh (tau) -cos (sigma)), but you can see that x**2 + y**2 = q**2
because sin^2(phi) + cos^2(phi) =1. This means that q is the radius of the toroid, not tau.

 

I forgot to mention that a and sigma and tau are just as in the Wikipedia link for bipolar coordinates (one of the links on the page that you referred to). There is a nice diagram there showing the distance between the Foci (a) and the coordinates sigma and tau. NB!
That article defines sigma and tau in the xy-plane, but your equations have the bipolar coordinates in the xz-plane.

 

No, your interpretation of sigma is not correct. I can give you a more detailed reply later but in the meantime, go to this link:
http://en.wikipedia.org/wiki/Bipolar_coordinates
and look at the first two diagrams, relabeling the y-axis as the z-axis to agree with the toroidal coordinates that you referred to. Sigma is the angle subtended by chord of length 2a (I think I made a mistake in an earlier post: the foci lie on the x-axis at +a and -a and so are separated by 2a) at the circle of constant sigma passing through the two foci.

 

Spelling it out.

Twinbee:
You are interested in toroidal coordinates in 3D, (sigma,tau,phi). These are
generated from bipolar coordinates (sigma,tau) in the 2D xz-plane by rotating
about the z-axis (i.e. in the xy-plane) by the angle phi. You want to start with some point S(x,y,z)
and you want its toroidal coordinates S(sigma,tau,phi). Do this in steps:
1) Viewing the z-axis as the polar axis, write S in terms of cylindrical
coordinates S(q,phi,z), where x = q cos(phi) and y = q sin(phi) so that
q² = x² + y² and
phi = Tan^-1(y/x).
2) This means that the point S can be obtained by rotating a point in the xz-plane,
P(q,0,z), through the angle phi about the z-axis. We
have now reduced the problem to finding the bipolar coordinates of
P(q,0,z).
3) You now have to choose the foci of the bipolar coordinates. Let the
first of these be F₁ with coordinates (-a,0,0) and the
second be F₂ at (+a,0,0). The points F₁PF₂
are the vertices of a triangle in the xz-plane.
4) the length of the line PF₁ is d₁ and that of line
PF₂ is d₂:
d₁² = (q+a)² + z²
d₂² = (q-a)² + z²
5) The coordinate tau is now obtained from its definition:
tau = log(d₁/d₂).
6) The coordinate sigma is computed from the law of cosines:
cos(sigma) = -(4a²-d₁²-d₂²)/(2d₁d₂)

That's it.
Now you should check this result. Take a point (x,y,z) and compute
(sigma,tau,phi) from the above; then, insert the result in the formula
from Wikipedia to get the original (x,y,z) back again. In this way, you
can find any mistakes I have made.

 

Sorry, but your diagram is completely incorrect. Please go to the link
http://en.wikipedia.org/wiki/Bipolar_coordinates
The first diagram you see displays sets of circles that intersect at right angles. The ones that pass through the two foci are curves of constant sigma, the others are curves of constant tau. Note that you should relabel the Y-axis as the z-axis in order to agree with the equations in the Wikipedia article that you referred to. The second diagram shows a red circle passing through the two foci (F1 at (-a,0) and F2 at (a,0)) and the point P(q,0,z). Sigma is clearly marked here. It is the angle subtended by the chord F1F2 at any point on the circle. The distances d1 and d2 from the point P to each of the foci are also shown. tau is log(d1/d2) BY DEFINITION. (Note that tau is not any old number but is determined from x,y,z and the position of the foci.) The point (x,y,z) is obtained by rotating the point P through an angle phi about the z-axis (i.e. in the plane Z=z). So, once you have chosen the parameter "a" to fit your problem, then the toroidal coordinates of any point (x,y,z) may be obtained from the program:

phi = ATAN2(y/x)
q = SQRT(x**2 + y**2)
d1 = SQRT((q+a)**2 + z**2)
d2 = SQRT((q-a)**2 + z**2)
tau = LOG(d1/d2)
sigma = ACOS( (d1**2 + d2**2 - 4*a**2)/(2*d1*d2) )

If you wish to improve on the efficiency of this program, be my guest. There are many ways to skin a cat.

 

No, it has nothing to do with the labels or their names. I am afraid your whole conception about the way the coordinate system looks like is mistaken. Your diagram looks like a ring of baloney (which is a torus, granted) that is sliced
through. Your angle sigma seems to be associated with just one end of the sliced-through torus, whereas sigma is supposed to have something to do with both of the circles of the sliced-through torus. Actually, the picture in 3D of these coordinates is more complicated than a torus. In the first picture of the bipolar diagram, consider just one of the circles through the foci (i.e. the circle of constant sigma) and one of the circles of constant tau about a focal point. Now grab the vertical axis between your thumb and index finger and twirl it about the vertical axis as you would spin a top. The circle through the foci will then sweep out a sphere in 3D, while the other circle will sweep out a torus around the vertical axis. Do you see that?

Your original post referred to a set of equations relating Cartesian coordinates to toroidal coordinates. These equations define everything and they are all I have to go on! There is no room for "intuition" anymore. Now if you wish to define some other kind of coordinates, that is up to you. You are free to do what you wish but then you have to tell me what they are!

It may be none of my business but what do you want these coordinates for? Do you have some problem or project where the "ring of baloney" would seem
appropriate? Then it may well be that you in fact need some coordinates other than the ones described by the equations in your original post! Mull it over.

 

OK, now I know what you are aiming for, but what role does the torus play in this?

I assume that (x,y,z) are the Cartesian coordinates. a and b are the major and minor radii of the torus, respectively; A is the angle for the rotation in the xy-plane; B is the angle for a rotation in the xz-plane. These equations tell you the Cartesian coordinates (x,y,z) of any point on the torus with fixed parameters a and b and the two coordinates A and B.
This is not a relation between two coordinate systems. Why? Because the (x,y,z) can describe any point in 3D space, whereas A and B describe only those points
on the surface of a particular torus characterised by a and b.

How are you going to specify four things (a,b,A,B) given only three coordinates (x,y,z)? Suppose you choose some value for b. Then, from z you might be able to calculate B = arcsin(z/b). Then you might be able to get a from
(x² + y²). Finally A=arctan(y/x). But this will be to no avail because a and B will depend upon the value that you chose for b; i.e. your "coordinate system" transformation is not unique.

In summary: either you wish to express points on the surface of a particular torus in terms of Cartesian coordinates, in which case you can only go one way (a,b,A,B ---> x,y,z) OR
you wish to express all points in 3D in terms of several different coordinate systems that can do this (e.g. the toroidal coordinates of the OP), in which case it is usually possible to obtain one set of coordinates from the other.

"Not I", said the cat. Life is too short to keep up with Wikipedia.

 

OK. How about the point (1,0,100)? Remember that b must not be greater
than "a" in a torus.

No, I haven't seen the broccoli stuff before. Nice. There is a publication about
3D fractals: Lasenby, J, R. Lasenby, A. Lasenby, and R. Wareham. "Higher
dimensional fractals in geometric algebra."
Technical Report CUED/F-INFENG/TR.556, Cambridge University
Engineering Department, 2006. Might be interesting for you.