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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

**to do my calculations.**

*Mathematica*