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.
Example 4.10
Let us show how to use the spherical coordinates in .
Let us check the ranges of variables .
Let us note that that the spherical coordinates in manifolds.Sphere
are defined on an open subset of the sphere, they do not cover the whole sphere. To cover the entire sphere with charts, we will use the stereographic coordinates instead.
Check the Jacobian of the embedding into ,
and the 2x2 minors of this matrix.
The sum of squares of 2x2 minors allows us to check the rank of the Jacobian matrix.
so the rank is 2 if .
If we are not interested in all details, we can check the rank of Jacobian using the command:
As we can see spherical coordinates define transformation with Jacobian of rank 2 on the subset A of the sphere defined by .
Example 4.11
Let us plot the coordinate lines of the spherical coordinates.
Example 4.12
We can restrict the ranges of parameters (for example to ):
Example 4.13
We can investigate and its parts without manifolds.Sphere
, but the transition maps and the corresponding restrictions must be defined by the user in that case.
We have to define the manifolds, charts and the embedding .
We can make previous plots without manifolds.Sphere
setup.
Stereographic coordinates in
Stereographic projection of from the North pole
Consider the line which passes through the North pole , the point on the sphere and intersects the plane Oxy at the point . The functions define the stereographic projection from the North pole. Using the rotations around z-axis and appropriate similar triangles in and planes we can see that (as in the case of ) so the stereographic projection can be described by Solving this system combined with the equation of the unit sphere: with respect to :
we obtain the transformation
Stereographic projection of from the South pole
Consider the line which passes through the South pole , the point on the sphere and intersects the plane Oxy at the point . The functions define a stereographic projection from the South pole. Using the rotations around z-axis and appropriate similar triangles in and planes we can see that (as in the case of ) so the stereographic projection from the South pole can be described by Solving this system combined with the equation of the unit sphere: with respect to :
we obtain the inverse transformation
Example 4.14
As we mentioned in the case of both the stereographic projections from North and South poles are predefined in manifolds.Sphere
.
Example 4.15
To show graphically how the projection from the North pole acts, let us extract an appropriate part of definition from the previous cell.
In the figure below we show the arcs on the sphere, which correspond to the lines on the plane (for the stereographic projection from the North pole).
Example 4.16
Now let us do the same for the South pole.
In the figure below we show the arcs on the sphere, which correspond to the lines on the plane (for the stereographic projection from the South pole).
Transition map
From the relations and it follows , so the transition map is of the form
To calculate the fraction let us note that where Accordingly Consequently the transition map has the form
Example 4.17
Transition maps from one stereographic projection to the second one in SageMath Manifolds
:
SageMath
can perform the corresponding calculation (with the help of Maxima).
The inverse transition map has the form
Both transformations are smooth provided and are nonzero.
Example 4.18
In fact, both transition maps are predefined in manifolds.Sphere
.
Example 4.19
The transitions from stereoN to stereoS and the inverse can be also obtained as compositions.
Example 4.20
Knowing the transition maps we are ready to an independent of manifolds.Sphere
definition of as a manifold with two maps.
What's next?
Take a look at the notebook Spheres and spherical coordinates in higher dimensions.