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Path: blob/main/04Manifold_Spheres.ipynb
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4. Spheres as manifolds
This notebook is part of the Introduction to manifolds in SageMath by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).
One dimensional sphere
Example 4.1
In manifolds.Sphere
in the one-dimensional case the default coordinates are the spherical (polar) ones.
The spherical coordinates are predefined in manifolds.Sphere
but we want to have a shorter name.
On the map is not one-to one, so not homeomorphic.
To represent graphically the sphere we need the ambient space and the mapping .
Restricting the coordinate to an open interval contained in we obtain a homeomorphic map (but the image is a proper subset of the sphere).
Example 4.2
can be defined as a manifold with two-element atlas. Note that in this new example we do not use the manifolds.Sphere
command!
On the intersection of two "new" charts are defined, they are just restrictions of the "old" maps to .
If we need to use in calculations we can define it:
To make the plot in we need to define an ambient space and the embedding.
Stereographic coordinates on
Example 4.3
In manifold.Sphere
not only the Cartesian and polar but also the stereographic coordinates are predefined.
Stereographic projection of from the North pole
Consider the line which passes through the North pole , the point on the circle and intersects the axis Ox at the point . The function defines the stereographic projection. Three points: North pole, the point on the circle and the point are on the same line. Since the right triangle with the hypotenuse joining (0,1), (x,y) and the right triangle with the hypotenuse joining (0,1),(u,0) are similar, we have and consequently . Solving the system of equations with respect to :
we obtain the relations
Now consider the South pole case.
Stereographic projection of from the South pole
Since the right triangle with the hypotenuse joining (0,-1), (x,y) and the right triangle with the hypotenuse joining (0,-1),(u',0) are similar we have ( is negative now) and consequently . Solving the system of equations with respect to :
we obtain the relations
Example 4.4
As we mentioned, both the stereographic projections from North and South poles are predefined in manifolds.Sphere
.
Example 4.5
To show graphically how the projection from the North pole acts, let us extract an appropriate part of the definition from the previous cell.
Let us plot the set of points corresponding to .
Thus the points of the grey arc in the figure are projected onto the interval (-10,10).
Example 4.6
Now let us extract the definition of the projection from the South pole,
and check which points of the circle are projected onto (-10,10):
Transition map from u coordinate to u' coordinate
Using the relations and we obtain Replacing by we get Accordingly so the transition map from coordinate to coordinate has the form It is smooth if
Here is SageMath solution (eliminate x,y variables from equations of the first line in the previous computations):
The answer means that the bracket vanishes. Since , the expression in the bracket vanishes if .
Example 4.7
In SageMath Manifolds
the transition from one coordinate system to the other can be defined as follows:
In some cases, the definition of the inverse transition can be left to the SageMath Manifolds
.
As we can see the transition map and its inverse are smooth.
Some transition maps are predefined in manifolds.Sphere
.
Example 4.8
Let us check that the transitions from the previous example are predefined in manifolds.Sphere
.
We can check that both transitions can be obtained as compositions of other transitions.
Example 4.9
Knowing the transition maps we are ready to define as a manifold with two maps.
Note that this time, our calculations are independent of manifolds.Sphere
.
Two dimensional sphere
Spherical coordinates in
In manifolds.Sphere
in two-dimensional case the default coordinates are the spherical ones.