Is there an explicit conformal map from ellipsoids to the plane or from ellipsoids to spheres?

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I just stumbled upon conformal maps (never heard about them before) by accident. And it seems I may be able to solve a problem I have been working on with them.

For this I need to find an explicit conformal map from an arbitrary ellipsoid to either a plane or a sphere, however merely searching the web for such a map isn't returning anything useful.

I am wondering if anyone knows of an explicit, bijective, conformal map from an arbitrary ellipsoid to either of the described manifolds.