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

\[\iota \colon \mathbb{S}^n_r \to E^{n+1}\]

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

\[\iota(\mathbb{S}^n_r(c)) = \left\{ p \in E^{n+1} : \lVert x(p) \rVert = r \right\}.\]

This corresponds to the standard \(n\)-sphere of radius \(r\) centered at \(c\).

AUTHORS:

  • Michael Jung (2020): initial version

REFERENCES:

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: h = S2_r.induced_metric()
sage: h.display()
gamma = 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: h 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)))

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: h = S1c.induced_metric()
sage: h.display()
gamma = dchi⊗dchi
sage: eps = h.volume_form()
sage: eps.display()
eps_gamma = -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 sphere

  • radius – (default: 1) positive number that states the radius of the sphere

  • name – (default: None) string; name (symbol) given to the sphere; if None, 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; if None, a new instance of Euclidean space is created

  • center – (default: None) the barycenter of the sphere as point of the ambient Euclidean space; if None the barycenter is set to the origin of the ambient space’s standard Cartesian coordinates

  • latex_name – (default: None) string; LaTeX symbol to denote the space; if None, 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

  • names – (default: None) must be a tuple containing the coordinate symbols (this guarantees the shortcut operator <,> to function); if None, 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 (without unique_tag, the UniqueRepresentation behavior inherited from PseudoRiemannianManifold 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 coordinates

  • names – (default: None) must be a tuple containing the coordinate symbols for the first chart in the list; if None, 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 and q on self.

INPUT:

  • p – an element of self

  • q – an element of self

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))