Spheres smoothly embedded in Euclidean Space¶
Let \(E^{n+1}\) be a Euclidean space of dimension \(n+1\) and \(c \in E^{n+1}\). An \(n\)-sphere with radius \(r\) and centered at \(c\), usually denoted by \(\mathbb{S}^n_r(c)\), smoothly embedded in the Euclidean space \(E^{n+1}\) is an \(n\)-dimensional smooth manifold together with a smooth embedding
whose image consists of all points having the same Euclidean distance to the fixed point \(c\). If we choose Cartesian coordinates \((x_1, \ldots, x_{n+1})\) on \(E^{n+1}\) with \(x(c)=0\) then the above translates to
This corresponds to the standard \(n\)-sphere of radius \(r\) centered at \(c\).
AUTHORS:
Michael Jung (2020): initial version
REFERENCES:
M. Berger: Geometry I&II [Ber1987], [Ber1987a]
J. Lee: Introduction to Smooth Manifolds [Lee2013]
EXAMPLES:
We start by defining a 2-sphere of unspecified radius \(r\):
sage: r = var('r')
sage: S2_r = manifolds.Sphere(2, radius=r); S2_r
2-sphere S^2_r of radius r smoothly embedded in the Euclidean space E^3
The embedding \(\iota\) is constructed from scratch and can be returned by the following command:
sage: i = S2_r.embedding(); i
Differentiable map iota from the 2-sphere S^2_r of radius r smoothly
embedded in the Euclidean space E^3 to the Euclidean space E^3
sage: i.display()
iota: S^2_r → E^3
on A: (theta, phi) ↦ (x, y, z) = (r*cos(phi)*sin(theta),
r*sin(phi)*sin(theta),
r*cos(theta))
As a submanifold of a Riemannian manifold, namely the Euclidean space, the 2-sphere admits an induced metric:
sage: g = S2_r.induced_metric()
sage: g.display()
g = r^2 dtheta⊗dtheta + r^2*sin(theta)^2 dphi⊗dphi
The induced metric is also known as the first fundamental form (see
first_fundamental_form()
):
sage: g is S2_r.first_fundamental_form()
True
The second fundamental form encodes the extrinsic curvature of the
2-sphere as hypersurface of Euclidean space (see
second_fundamental_form()
):
sage: K = S2_r.second_fundamental_form(); K
Field of symmetric bilinear forms K on the 2-sphere S^2_r of radius r
smoothly embedded in the Euclidean space E^3
sage: K.display()
K = r dtheta⊗dtheta + r*sin(theta)^2 dphi⊗dphi
One quantity that can be derived from the second fundamental form is the Gaussian curvature:
sage: K = S2_r.gauss_curvature()
sage: K.display()
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:
sage: S2_r.stereographic_coordinates()
Chart (S^2_r-{NP}, (y1, y2))
To get all charts corresponding to stereographic coordinates, we can use the
coordinate_charts()
:
sage: stereoN, stereoS = S2_r.coordinate_charts('stereographic')
sage: stereoN, stereoS
(Chart (S^2_r-{NP}, (y1, y2)), Chart (S^2_r-{SP}, (yp1, yp2)))
See also
See stereographic_coordinates()
and spherical_coordinates()
for details.
Note
Notice that the derived quantities such as the embedding as well as the first and second fundamental forms must be computed from scratch again when new coordinates have been initialized. That makes the usage of previously declared objects obsolete.
Consider now a 1-sphere with barycenter \((1,0)\) in Cartesian coordinates:
sage: E2 = EuclideanSpace(2)
sage: c = E2.point((1,0), name='c')
sage: S1c.<chi> = E2.sphere(center=c); S1c
1-sphere S^1(c) of radius 1 smoothly embedded in the Euclidean plane
E^2 centered at the Point c
sage: S1c.spherical_coordinates()
Chart (A, (chi,))
Get stereographic coordinates:
sage: stereoN, stereoS = S1c.coordinate_charts('stereographic')
sage: stereoN, stereoS
(Chart (S^1(c)-{NP}, (y1,)), Chart (S^1(c)-{SP}, (yp1,)))
The embedding takes now the following form in all coordinates:
sage: S1c.embedding().display()
iota: S^1(c) → E^2
on A: chi ↦ (x, y) = (cos(chi) + 1, sin(chi))
on S^1(c)-{NP}: y1 ↦ (x, y) = (2*y1/(y1^2 + 1) + 1, (y1^2 - 1)/(y1^2 + 1))
on S^1(c)-{SP}: yp1 ↦ (x, y) = (2*yp1/(yp1^2 + 1) + 1, -(yp1^2 - 1)/(yp1^2 + 1))
Since the sphere is a hypersurface, we can get a normal vector field by using
normal
:
sage: n = S1c.normal(); n
Vector field n along the 1-sphere S^1(c) of radius 1 smoothly embedded in
the Euclidean plane E^2 centered at the Point c with values on the
Euclidean plane E^2
sage: n.display()
n = -cos(chi) e_x - sin(chi) e_y
Notice that this is just one normal field with arbitrary direction, in this particular case \(n\) points inwards whereas \(-n\) points outwards. However, the vector field \(n\) is indeed non-vanishing and hence the sphere admits an orientation (as all spheres do):
sage: orient = S1c.orientation(); orient
[Coordinate frame (S^1(c)-{SP}, (∂/∂yp1)), Vector frame (S^1(c)-{NP}, (f_1))]
sage: f = orient[1]
sage: f[1].display()
f_1 = -∂/∂y1
Notice that the orientation is chosen is such a way that \((\iota_*(f_1), -n)\) is oriented in the ambient Euclidean space, i.e. the last entry is the normal vector field pointing outwards. Henceforth, the manifold admits a volume form:
sage: g = S1c.induced_metric()
sage: g.display()
g = dchi⊗dchi
sage: eps = g.volume_form()
sage: eps.display()
eps_g = -dchi
- class sage.manifolds.differentiable.examples.sphere.Sphere(n, radius=1, ambient_space=None, center=None, name=None, latex_name=None, coordinates='spherical', names=None, category=None, init_coord_methods=None, unique_tag=None)¶
Bases:
sage.manifolds.differentiable.pseudo_riemannian_submanifold.PseudoRiemannianSubmanifold
Sphere smoothly embedded in Euclidean Space.
An \(n\)-sphere of radius \(r\) is a smooth \(n\)-dimensional manifold smoothly embedded into \(E^{n+1}\), such that the embedding constitutes a standard \(n\)-sphere of radius \(r\) in that Euclidean space (possibly shifted by a point).
n
– positive integer representing dimension of the sphereradius
– (default:1
) positive number that states the radius of the spherename
– (default:None
) string; name (symbol) given to the sphere; ifNone
, the name will be set according to the input (see convention above)ambient_space
– (default:None
) Euclidean space in which the sphere should be embedded; ifNone
, a new instance of Euclidean space is createdcenter
– (default:None
) the barycenter of the sphere as point of the ambient Euclidean space; ifNone
the barycenter is set to the origin of the ambient space’s standard Cartesian coordinateslatex_name
– (default:None
) string; LaTeX symbol to denote the space; ifNone
, it will be set according to the input (see convention above)coordinates
– (default:'spherical'
) string describing the type of coordinates to be initialized at the sphere’s creation; allowed values are'spherical'
spherical coordinates (seespherical_coordinates()
))'stereographic'
stereographic coordinates given by the stereographic projection (seestereographic_coordinates()
)
names
– (default:None
) must be a tuple containing the coordinate symbols (this guarantees the shortcut operator<,>
to function); ifNone
, the usual conventions are used (see examples below for details)unique_tag
– (default:None
) tag used to force the construction of a new object when all the other arguments have been used previously (withoutunique_tag
, theUniqueRepresentation
behavior inherited fromPseudoRiemannianManifold
would return the previously constructed object corresponding to these arguments)
EXAMPLES:
A 2-sphere embedded in Euclidean space:
sage: S2 = manifolds.Sphere(2); S2 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3 sage: latex(S2) \mathbb{S}^{2}
The ambient Euclidean space is constructed incidentally:
sage: S2.ambient() Euclidean space E^3
Another call creates another sphere and hence another Euclidean space:
sage: S2 is manifolds.Sphere(2) False sage: S2.ambient() is manifolds.Sphere(2).ambient() False
By default, the barycenter is set to the coordinate origin of the standard Cartesian coordinates in the ambient Euclidean space:
sage: c = S2.center(); c Point on the Euclidean space E^3 sage: c.coord() (0, 0, 0)
Each \(n\)-sphere is a compact manifold and a complete metric space:
sage: S2.category() Join of Category of compact topological spaces and Category of smooth manifolds over Real Field with 53 bits of precision and Category of connected manifolds over Real Field with 53 bits of precision and Category of complete metric spaces
If not stated otherwise, each \(n\)-sphere is automatically endowed with spherical coordinates:
sage: S2.atlas() [Chart (A, (theta, phi))] sage: S2.default_chart() Chart (A, (theta, phi)) sage: spher = S2.spherical_coordinates() sage: spher is S2.default_chart() True
Notice that the spherical coordinates do not cover the whole sphere. To cover the entire sphere with charts, use stereographic coordinates instead:
sage: stereoN, stereoS = S2.coordinate_charts('stereographic') sage: stereoN, stereoS (Chart (S^2-{NP}, (y1, y2)), Chart (S^2-{SP}, (yp1, yp2))) sage: list(S2.open_covers()) [Set {S^2} of open subsets of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3, Set {S^2-{NP}, S^2-{SP}} of open subsets of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3]
Note
Keep in mind that the initialization process of stereographic coordinates and their transition maps is computational complex in higher dimensions. Henceforth, high computation times are expected with increasing dimension.
- center()¶
Return the barycenter of
self
in the ambient Euclidean space.EXAMPLES:
2-sphere embedded in Euclidean space centered at \((1,2,3)\) in Cartesian coordinates:
sage: E3 = EuclideanSpace(3) sage: c = E3.point((1,2,3), name='c') sage: S2c = manifolds.Sphere(2, ambient_space=E3, center=c); S2c 2-sphere S^2(c) of radius 1 smoothly embedded in the Euclidean space E^3 centered at the Point c sage: S2c.center() Point c on the Euclidean space E^3
We can see that the embedding is shifted accordingly:
sage: S2c.embedding().display() iota: S^2(c) → E^3 on A: (theta, phi) ↦ (x, y, z) = (cos(phi)*sin(theta) + 1, sin(phi)*sin(theta) + 2, cos(theta) + 3)
- coordinate_charts(coord_name, names=None)¶
Return a list of all charts belonging to the coordinates
coord_name
.INPUT:
coord_name
– string describing the type of coordinatesnames
– (default:None
) must be a tuple containing the coordinate symbols for the first chart in the list; ifNone
, the standard convention is used
EXAMPLES:
Spherical coordinates on \(S^1\):
sage: S1 = manifolds.Sphere(1) sage: S1.coordinate_charts('spherical') [Chart (A, (phi,))]
Stereographic coordinates on \(S^1\):
sage: stereo_charts = S1.coordinate_charts('stereographic', names=['a']) sage: stereo_charts [Chart (S^1-{NP}, (a,)), Chart (S^1-{SP}, (ap,))]
- dist(p, q)¶
Return the great circle distance between the points
p
andq
onself
.INPUT:
p
– an element ofself
q
– an element ofself
OUTPUT:
the great circle distance \(d(p, q)\) on
self
The great circle distance \(d(p, q)\) of the points \(p, q \in \mathbb{S}^n_r(c)\) is the length of the shortest great circle segment on \(\mathbb{S}^n_r(c)\) that joins \(p\) and \(q\). If we choose Cartesian coordinates \((x_1, \ldots, x_{n+1})\) of the ambient Euclidean space such that the center lies in the coordinate origin, i.e. \(x(c)=0\), the great circle distance can be expressed in terms of the following formula:
\[d(p,q) = r \, \arccos\left(\frac{x(\iota(p)) \cdot x(\iota(q))}{r^2}\right).\]EXAMPLES:
Define a 2-sphere with unspecified radius:
sage: r = var('r') sage: S2_r = manifolds.Sphere(2, radius=r); S2_r 2-sphere S^2_r of radius r smoothly embedded in the Euclidean space E^3
Given two antipodal points in spherical coordinates:
sage: p = S2_r.point((pi/2, pi/2), name='p'); p Point p on the 2-sphere S^2_r of radius r smoothly embedded in the Euclidean space E^3 sage: q = S2_r.point((pi/2, -pi/2), name='q'); q Point q on the 2-sphere S^2_r of radius r smoothly embedded in the Euclidean space E^3
The distance is determined as the length of the half great circle:
sage: S2_r.dist(p, q) pi*r
- minimal_triangulation()¶
Return the minimal triangulation of
self
as a simplicial complex.EXAMPLES:
Minimal triangulation of the 2-sphere:
sage: S2 = manifolds.Sphere(2) sage: S = S2.minimal_triangulation(); S Minimal triangulation of the 2-sphere
The Euler characteristic of a 2-sphere:
sage: S.euler_characteristic() 2
- radius()¶
Return the radius of
self
.EXAMPLES:
3-sphere with radius 3:
sage: S3_2 = manifolds.Sphere(3, radius=2); S3_2 3-sphere S^3_2 of radius 2 smoothly embedded in the 4-dimensional Euclidean space E^4 sage: S3_2.radius() 2
2-sphere with unspecified radius:
sage: r = var('r') sage: S2_r = manifolds.Sphere(3, radius=r); S2_r 3-sphere S^3_r of radius r smoothly embedded in the 4-dimensional Euclidean space E^4 sage: S2_r.radius() r
- spherical_coordinates(names=None)¶
Return the spherical coordinates of
self
.INPUT:
names
– (default:None
) must be a tuple containing the coordinate symbols (this guarantees the usage of the shortcut operator<,>
)
OUTPUT:
the chart of spherical coordinates, as an instance of
RealDiffChart
Let \(\mathbb{S}^n_r(c)\) be an \(n\)-sphere of radius \(r\) smoothly embedded in the Euclidean space \(E^{n+1}\) centered at \(c \in E^{n+1}\). We say that \((\varphi_1, \ldots, \varphi_n)\) define spherical coordinates on the open subset \(A \subset \mathbb{S}^n_r(c)\) for the Cartesian coordinates \((x_1, \ldots, x_{n+1})\) on \(E^{n+1}\) (not necessarily centered at \(c\)) if
\[\begin{split}\begin{aligned} \left. x_1 \circ \iota \right|_{A} &= r \cos(\varphi_n)\sin( \varphi_{n-1}) \cdots \sin(\varphi_1) + x_1(c), \\ \left. x_1 \circ \iota \right|_{A} &= r \sin(\varphi_n)\sin( \varphi_{n-1}) \cdots \sin(\varphi_1) + x_1(c), \\ \left. x_2 \circ \iota \right|_{A} &= r \cos(\varphi_{ n-1})\sin(\varphi_{n-2}) \cdots \sin(\varphi_1) + x_2(c), \\ \left. x_3 \circ \iota \right|_{A} &= r \cos(\varphi_{ n-2})\sin(\varphi_{n-3}) \cdots \sin(\varphi_1) + x_3(c), \\ \vdots & \\ \left. x_{n+1} \circ \iota \right|_{A} &= r \cos(\varphi_1) + x_{n+1}(c), \end{aligned}\end{split}\]where \(\varphi_i\) has range \((0, \pi)\) for \(i=1, \ldots, n-1\) and \(\varphi_n\) lies in \((-\pi, \pi)\). Notice that the above expressions together with the ranges of the \(\varphi_i\) fully determine the open set \(A\).
Note
Notice that our convention slightly differs from the one given on the Wikipedia article N-sphere#Spherical_coordinates. The definition above ensures that the conventions for the most common cases \(n=1\) and \(n=2\) are maintained.
EXAMPLES:
The spherical coordinates on a 2-sphere follow the common conventions:
sage: S2 = manifolds.Sphere(2) sage: spher = S2.spherical_coordinates(); spher Chart (A, (theta, phi))
The coordinate range of spherical coordinates:
sage: spher.coord_range() theta: (0, pi); phi: [-pi, pi] (periodic)
Spherical coordinates do not cover the 2-sphere entirely:
sage: A = spher.domain(); A Open subset A of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3
The embedding of a 2-sphere in Euclidean space via spherical coordinates:
sage: S2.embedding().display() iota: S^2 → E^3 on A: (theta, phi) ↦ (x, y, z) = (cos(phi)*sin(theta), sin(phi)*sin(theta), cos(theta))
Now, consider spherical coordinates on a 3-sphere:
sage: S3 = manifolds.Sphere(3) sage: spher = S3.spherical_coordinates(); spher Chart (A, (chi, theta, phi)) sage: S3.embedding().display() iota: S^3 → E^4 on A: (chi, theta, phi) ↦ (x1, x2, x3, x4) = (cos(phi)*sin(chi)*sin(theta), sin(chi)*sin(phi)*sin(theta), cos(theta)*sin(chi), cos(chi))
By convention, the last coordinate is periodic:
sage: spher.coord_range() chi: (0, pi); theta: (0, pi); phi: [-pi, pi] (periodic)
- stereographic_coordinates(pole='north', names=None)¶
Return stereographic coordinates given by the stereographic projection of
self
w.r.t. to a given pole.INPUT:
pole
– (default:'north'
) the pole determining the stereographic projection; possible options are'north'
and'south'
names
– (default:None
) must be a tuple containing the coordinate symbols (this guarantees the usage of the shortcut operator<,>
)
OUTPUT:
the chart of stereographic coordinates w.r.t. to the given pole, as an instance of
RealDiffChart
Let \(\mathbb{S}^n_r(c)\) be an \(n\)-sphere of radius \(r\) smoothly embedded in the Euclidean space \(E^{n+1}\) centered at \(c \in E^{n+1}\). We denote the north pole of \(\mathbb{S}^n_r(c)\) by \(\mathrm{NP}\) and the south pole by \(\mathrm{SP}\). These poles are uniquely determined by the requirement
\[\begin{split}x(\iota(\mathrm{NP})) &= (0, \ldots, 0, r) + x(c), \\ x(\iota(\mathrm{SP})) &= (0, \ldots, 0, -r) + x(c).\end{split}\]The coordinates \((y_1, \ldots, y_n)\) (\((y'_1, \ldots, y'_n)\) respectively) define stereographic coordinates on \(\mathbb{S}^n_r(c)\) for the Cartesian coordinates \((x_1, \ldots, x_{n+1})\) on \(E^{n+1}\) if they arise from the stereographic projection from \(\iota(\mathrm{NP})\) (\(\iota(\mathrm{SP})\)) to the hypersurface \(x_n = x_n(c)\). In concrete formulas, this means:
\[\begin{split}\left. x \circ \iota \right|_{\mathbb{S}^n_r(c) \setminus \{ \mathrm{NP}\}} &= \left( \frac{2y_1r^2}{r^2+\sum^n_{i=1} y^2_i}, \ldots, \frac{2y_nr^2}{r^2+\sum^n_{i=1} y^2_i}, \frac{r\sum^n_{i=1} y^2_i-r^3}{r^2+\sum^n_{i=1} y^2_i} \right) + x(c), \\ \left. x \circ \iota \right|_{\mathbb{S}^n_r(c) \setminus \{ \mathrm{SP}\}} &= \left( \frac{2y'_1r^2}{r^2+\sum^n_{i=1} y'^2_i}, \ldots, \frac{2y'_nr^2}{r^2+\sum^n_{i=1} y'^2_i}, \frac{r^3 - r\sum^n_{i=1} y'^2_i}{r^2+\sum^n_{i=1} y'^2_i} \right) + x(c).\end{split}\]EXAMPLES:
Initialize a 1-sphere centered at \((1,0)\) in the Euclidean plane using the shortcut operator:
sage: E2 = EuclideanSpace(2) sage: c = E2.point((1,0), name='c') sage: S1.<a> = E2.sphere(center=c, coordinates='stereographic'); S1 1-sphere S^1(c) of radius 1 smoothly embedded in the Euclidean plane E^2 centered at the Point c
By default, the shortcut variables belong to the stereographic projection from the north pole:
sage: S1.coordinate_charts('stereographic') [Chart (S^1(c)-{NP}, (a,)), Chart (S^1(c)-{SP}, (ap,))] sage: S1.embedding().display() iota: S^1(c) → E^2 on S^1(c)-{NP}: a ↦ (x, y) = (2*a/(a^2 + 1) + 1, (a^2 - 1)/(a^2 + 1)) on S^1(c)-{SP}: ap ↦ (x, y) = (2*ap/(ap^2 + 1) + 1, -(ap^2 - 1)/(ap^2 + 1))
Initialize a 2-sphere from scratch:
sage: S2 = manifolds.Sphere(2) sage: S2.atlas() [Chart (A, (theta, phi))]
In the previous block, the stereographic coordinates have not been initialized. This happens subsequently with the invocation of
stereographic_coordinates
:sage: stereoS.<u,v> = S2.stereographic_coordinates(pole='south') sage: S2.coordinate_charts('stereographic') [Chart (S^2-{NP}, (up, vp)), Chart (S^2-{SP}, (u, v))]
If not specified by the user, the default coordinate names are given by \((y_1, \ldots, y_n)\) and \((y'_1, \ldots, y'_n)\) respectively:
sage: S3 = manifolds.Sphere(3, coordinates='stereographic') sage: S3.stereographic_coordinates(pole='north') Chart (S^3-{NP}, (y1, y2, y3)) sage: S3.stereographic_coordinates(pole='south') Chart (S^3-{SP}, (yp1, yp2, yp3))