From 4358faffec8b28e8fe704558a2bd19b8400ed087 Mon Sep 17 00:00:00 2001 From: Michael Jung Date: Sun, 14 Feb 2021 14:29:05 +0100 Subject: [PATCH] Trac #31324: documentation on atlas behavior of spherical coordinates --- .../differentiable/examples/sphere.py | 18 ++++++++++++++---- 1 file changed, 14 insertions(+), 4 deletions(-) diff --git a/src/sage/manifolds/differentiable/examples/sphere.py b/src/sage/manifolds/differentiable/examples/sphere.py index 67b40872c7b..e9086b2850a 100644 --- a/src/sage/manifolds/differentiable/examples/sphere.py +++ b/src/sage/manifolds/differentiable/examples/sphere.py @@ -80,14 +80,15 @@ 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') @@ -810,6 +811,15 @@ def spherical_coordinates(self, names=None): above ensures that the conventions for the most common cases `n=1` and `n=2` are maintained. + .. WARNING:: + + Even though spherical coordinates are internally defined as + periodic, keep noticed that they are, + in a mathematical sense, only defined on + the open subset `A` and do *not* cover the whole sphere + `\mathbb{S}^n_r(c)`. To get charts leading to a fully defined atlas, + use :meth:`Sphere.stereographic_coordinates` instead. + EXAMPLES: The spherical coordinates on a 2-sphere follow the common conventions::