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any tips on finding the inverse of the following map:

$$(\theta,\phi)\mapsto ((2+\cos\theta)\cos(\phi),(2+\cos\theta)\sin(\phi),\sin\theta)$$

From doing it with cartesian :

One of the maps is: I get $\phi^{\pm}(x_1,x_2,x_3)=(x_1,x_2)$ and

$$(\phi^{\pm})^{-1}(x_1,x_2) = \left(x_1,x_2,\pm\sqrt{1-\left(\sqrt{x_1^2+x_2^2}-2\right)^2}\right)$$

I will add more as I go.

thanks

Michael Hardy
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TKM
  • 2,485

1 Answers1

-1

This might help

ParametricTorus[u_, v_, a_, b_] := {(a Cos[u] + b) Cos[v], (a Cos[u] + b) Sin[v], a Sin[u]};

InverseParametricTorus[x_, y_, z_, a_, b_] := With[{v = Pi/2 - ArcTan[y, x], v2 = ArcTan[y, x] + 2 Pi}, With[{u = Pi/2 - ArcTan[z, (x/Cos[v] - b)]}, {UnitStep[u]u + (1 - UnitStep[u])(u + 2 Pi), UnitStep[v]v + (1 - UnitStep[v])(v + 2 Pi)}]];

kujoe_b
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