Uniform distribution on a toroidal surface

  • Thread starter Barnak
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I'm not sure this is the right thread to post my problem :

I'm trying to define a uniform distribution on the toroidal surface associated to a dipolar magnetic field (or electric). More specifically, the surface (in 3D euclidian space) is parametrised as this, using the usual polar coordinates :

[itex]x(\theta, \phi) = \sin^3{\theta} \; \cos{\phi},[/itex]
[itex]y(\theta, \phi) = \sin^3{\theta} \; \sin{\phi},[/itex]
[itex]z(\theta) = \sin^2{\theta} \; \cos{\theta}.[/itex]

The surface element is this :

[itex]dS(\theta, \phi) = \sin^7{\theta} \; d\theta \; d\phi.[/itex]

For a simple sphere, we get

[itex]dS_{sphere}(\theta, \phi) = \sin{\theta} \; d\theta \; d\phi = du \; d\phi,[/itex]

where [itex]u = \cos \theta[/itex] is the natural variable to define the uniform distribution on the sphere.

In the case of my toroidal surface defined above, the "natural" variable (if I'm not doing a mistake) is really complicated :

[itex]u = \cos{\theta} - \cos^3{\theta} + \tfrac{3}{5} \cos^5{\theta} - \tfrac{1}{7}\cos^7{\theta},[/itex]

so [itex]dS(u, \phi) = du \; d\phi[/itex]. This is the variable I should use to define an uniform distribution of points on the surface.
However, how should I define the three parametric coordinates [itex]x(u, \phi)[/itex], [itex]y(u, \phi)[/itex], [itex]z(u)[/itex] ? I'm unable to invert the function above to give [itex]\cos \theta = f(u) = ?[/itex]

Help please !

I'm using Mathematica to do my calculations.
 

fresh_42

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##0=-u + \cos \theta \cdot \left( 1 - (\cos^2 \theta) + \frac{3}{5} (\cos^2 \theta)^2 -\frac{1}{7}(\cos^2 \theta)^3\right)##
##= -u + \cos \theta \cdot (1-v+\frac{3}{5}v^2-\frac{1}{7}v^3)## which can basically be solved for ##v=\cos^2 \theta## and then for ##\cos \theta##.
 

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