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The Sphere
The optimal shape, a round sphere.Stereographic Parameterization
In the lower window you can see another sphere, but this time parameterized with stereographic coordinates. The advantage of this parameterization is that there is only one point, that is not differentiable.While in the geographic parameterization, shown in the upper window, the two poles and the line of longitude 0 are not differentiable.
The price of course is, that for getting the whole sphere it is necessary to take a domain from minus infinity to plus infinity in both coordinates.
