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Trac #31324: documentation on atlas behavior of spherical coordinates
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mjungmath committed Feb 14, 2021
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16 changes: 12 additions & 4 deletions src/sage/manifolds/differentiable/examples/sphere.py
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S^2_r --> R
on A: (theta, phi) |--> r^(-2)
As we have seen, spherical coordinates are initialized by default. To
initialize stereographic coordinates retrospectively, we can use the following
command::
By default, spheres are initialized with spherical coordinates. Even though
spherical coordinates are internally defined periodically, they do not
constitute a full atlas in the mathematical sense. To retrospectively
initialize stereographic coordinates, we can use the following command::
sage: S2_r.stereographic_coordinates()
Chart (S^2_r-{NP}, (y1, y2))
To get all charts corresponding to stereographic coordinates, we can use the
To return the full atlas constituted by stereographic coordinates, we can use the
:meth:`~sage.manifolds.differentiable.examples.sphere.Sphere.coordinate_charts`::
sage: stereoN, stereoS = S2_r.coordinate_charts('stereographic')
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above ensures that the conventions for the most common cases
`n=1` and `n=2` are maintained.
.. WARNING::
Keep noticed that these spherical coordinates are only defined on
the open subset `A` and do *not* cover the whole sphere
`\mathbb{S}^n_r(c)`. To get a fully defined atlas,
use :meth:`Sphere.stereographic_coordinates` instead.
EXAMPLES:
The spherical coordinates on a 2-sphere follow the common conventions::
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