05Manifold_Spher_Higher_Dim.ipynb

Kernel: SageMath 9.6

5. Spheres and spherical coordinates in higher dimensions

This notebook is part of the Introduction to manifolds in SageMath by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).

version()

'SageMath version 9.6, Release Date: 2022-05-15'

In the EuclideanSpace class, the transitions from spherical to Cartesian and Cartesian to spherical coordinates are predefined in the case $n\leq 3$ (SageMath 9.5).

Example 5.1

Let us demonstrate spherical coordinates for $n=3$.

E.<x1,x2,x3>=EuclideanSpace(3,"E")           # Euclidean space E^3
cart=E.cartesian_coordinates()               # Cartesian coordinates
spher.<r,th,ph> = E.spherical_coordinates()  # spherical coordinates
E.coord_change(spher, cart).display()     # transition spher -> cart 

x1 = r*cos(ph)*sin(th)
x2 = r*sin(ph)*sin(th)
x3 = r*cos(th)

E.coord_change(cart, spher).display()    # transition cart -> spher

r = sqrt(x1^2 + x2^2 + x3^2)
th = arctan2(sqrt(x1^2 + x2^2), x3)
ph = arctan2(x2, x1)

The manifolds.Sphere module allows for higher dimensions.

We can use predefined embeddings and transition functions on higher dimensional spheres.

Example 5.2

Let us demonstrate the case of 3-dimensional sphere in $E^4$.

%display latex
dim=3           # dimension of the sphere (not the Euclidean space)
Sph=manifolds.Sphere(dim)            # sphere S^3
spher = Sph.spherical_coordinates()  # spherical coordinates on S^3
Sph.embedding().disp()               # embedding S^3 -> E^4

$\displaystyle \begin{array}{llcl} \iota:& \mathbb{S}^{3} & \longrightarrow & \mathbb{E}^{4} \\ \mbox{on}\ A : & \left(\chi, \theta, \phi\right) & \longmapsto & \left({x_{1}}, {x_{2}}, {x_{3}}, {x_{4}}\right) = \left(\cos\left(\phi\right) \sin\left(\chi\right) \sin\left(\theta\right), \sin\left(\chi\right) \sin\left(\phi\right) \sin\left(\theta\right), \cos\left(\theta\right) \sin\left(\chi\right), \cos\left(\chi\right)\right) \end{array}$

The stereographic projections are also accessible in higher dimensions.

Example 5.3

Stereographic coordinates in $S^3$.

# continuation
%display latex
stereoN, stereoS = Sph.coordinate_charts('stereographic',
        names=['u','v','w'])      # stereographic projections in S^3
Sph.embedding().disp()            # show embeddings into E^4

$\displaystyle \begin{array}{llcl} \iota:& \mathbb{S}^{3} & \longrightarrow & \mathbb{E}^{4} \\ \mbox{on}\ A : & \left(\chi, \theta, \phi\right) & \longmapsto & \left({x_{1}}, {x_{2}}, {x_{3}}, {x_{4}}\right) = \left(\cos\left(\phi\right) \sin\left(\chi\right) \sin\left(\theta\right), \sin\left(\chi\right) \sin\left(\phi\right) \sin\left(\theta\right), \cos\left(\theta\right) \sin\left(\chi\right), \cos\left(\chi\right)\right) \\ \mbox{on}\ \mathbb{S}^{3}\setminus\{\mathrm{NP}\} : & \left(u, v, w\right) & \longmapsto & \left({x_{1}}, {x_{2}}, {x_{3}}, {x_{4}}\right) = \left(\frac{2 \, u}{u^{2} + v^{2} + w^{2} + 1}, \frac{2 \, v}{u^{2} + v^{2} + w^{2} + 1}, \frac{2 \, w}{u^{2} + v^{2} + w^{2} + 1}, \frac{u^{2} + v^{2} + w^{2} - 1}{u^{2} + v^{2} + w^{2} + 1}\right) \\ \mbox{on}\ \mathbb{S}^{3}\setminus\{\mathrm{SP}\} : & \left({u'}, {v'}, {w'}\right) & \longmapsto & \left({x_{1}}, {x_{2}}, {x_{3}}, {x_{4}}\right) = \left(\frac{2 \, {u'}}{{u'}^{2} + {v'}^{2} + {w'}^{2} + 1}, \frac{2 \, {v'}}{{u'}^{2} + {v'}^{2} + {w'}^{2} + 1}, \frac{2 \, {w'}}{{u'}^{2} + {v'}^{2} + {w'}^{2} + 1}, -\frac{{u'}^{2} + {v'}^{2} + {w'}^{2} - 1}{{u'}^{2} + {v'}^{2} + {w'}^{2} + 1}\right) \end{array}$

We have also the corresponding transitions in $S^3$ predefined.

A = spher.domain()               # domain of spherical coordinates
# intersection of domains of stereoN and stereoS coordinates:
W = Sph._stereoN_dom.intersection(Sph._stereoS_dom)
# intersection of domains of stereoN,stereoS and spher coordinates:
V=W.intersection(A) 
                                # transition spher -> stereoN:
F1=Sph.coord_change(spher.restrict(V), stereoN.restrict(V))
                                # transition stereoN -> spher:
F2=Sph.coord_change( stereoN.restrict(V),spher.restrict(V))
                                # transition spher -> stereoS 
F3=Sph.coord_change(spher.restrict(V), stereoS.restrict(V))
                                # transition stereoS -> spher
F4=Sph.coord_change( stereoS.restrict(V),spher.restrict(V))
                                # transition stereoN -> stereoS
F5=Sph.coord_change( stereoN.restrict(V),stereoS.restrict(V))
                                # transition stereoS -> stereoN
F6=Sph.coord_change( stereoS.restrict(V),stereoN.restrict(V))
F6.disp()

$\displaystyle \left\{\begin{array}{lcl} u & = & \frac{{u'}}{{u'}^{2} + {v'}^{2} + {w'}^{2}} \\ v & = & \frac{{v'}}{{u'}^{2} + {v'}^{2} + {w'}^{2}} \\ w & = & \frac{{w'}}{{u'}^{2} + {v'}^{2} + {w'}^{2}} \end{array}\right.$

F5.disp()

$\displaystyle \left\{\begin{array}{lcl} {u'} & = & \frac{u}{u^{2} + v^{2} + w^{2}} \\ {v'} & = & \frac{v}{u^{2} + v^{2} + w^{2}} \\ {w'} & = & \frac{w}{u^{2} + v^{2} + w^{2}} \end{array}\right.$

Example 5.4

Spherical coordinates in $R^4$.

We can use manifolds.Sphere to obtain the transition from spherical to Cartesian coordinates in $R^4$.

%display plain # to obtain output which can be re-used in the code
dim=3                              # dimension of the sphere
r=var('r')                         # symbolic variable
Sph=manifolds.Sphere(dim,radius=r) # 3-dim sphere with radius r
Phi=Sph.embedding()                # embedding into E^4
E=Sph.ambient()                    # ambient space E^4
fun=Phi.coord_functions()          # embedding functions
fun.expr()                         # show embedding functions

(r*cos(phi)*sin(chi)*sin(theta),
 r*sin(chi)*sin(phi)*sin(theta),
 r*cos(theta)*sin(chi),
 r*cos(chi))

The result can be used to define the transition spherU -> cartU in $R^4$.

R4=Manifold(4,'R^4',start_index=1) # manifold R^4
cart.<x1,x2,x3,x4>=R4.chart()      # Cartesian coordinates in R^4
U=R4.open_subset('U',              # open subset of R^4 with half-
  coord_def={cart: (x2!=0, x1<0)}) # plane {x2 = 0, x1 ≥ 0} excluded
cartU=cart.restrict(U)             # restrict cart to U
                                   # spherical coordinates on U, use
                                   # names from the previous output!
spherU.<r,chi,theta,phi>=U.chart(r'r:(0,+oo) chi:(0,pi):\chi\
    theta:(0,pi):\theta  phi:(-pi,pi):periodic:\phi')
                                   # transition spherU -> cartU
spher_to_cart = spherU.transition_map(cartU,fun.expr())
spher_to_cart.disp()               # show transition

x1 = r*cos(phi)*sin(chi)*sin(theta)
x2 = r*sin(chi)*sin(phi)*sin(theta)
x3 = r*cos(theta)*sin(chi)
x4 = r*cos(chi)

# spher_to_cart.inverse() does not work

Inverse transition can be defined as follows:

fun=(sqrt(x1^2+x2^2+x3^2+x4^2),        # functions defining
     atan2(sqrt(x1^2+x2^2+x3^2),x4),   # inverse transition
     atan2(sqrt(x1^2+x2^2),x3),        # spher -> cart
     atan2(x2,x1))
spher_to_cart.set_inverse(*fun)        # set inverse trans.

Check of the inverse coordinate transformation:
  r == r  *passed*
  chi == arctan2(r*sin(chi), r*cos(chi))  **failed**
  theta == arctan2(r*sin(chi)*sin(theta), r*cos(theta)*sin(chi))  **failed**
  phi == arctan2(r*sin(chi)*sin(phi)*sin(theta), r*cos(phi)*sin(chi)*sin(theta))  **failed**
  x1 == x1  *passed*
  x2 == x2  *passed*
  x3 == x3  *passed*
  x4 == x4  *passed*
NB: a failed report can reflect a mere lack of simplification.

The failed computations reflect a lack of simplification in SageMath and can be checked by hand (https://en.wikipedia.org/wiki/Atan2)

Example 5.5

Spherical coordinates in $R^5$.

%display plain                       # We don't want the Latex output
dim=4                                # dimension of the sphere
r=var('r')                           # symbolic variable
Sph=manifolds.Sphere(dim,radius=r)   # 4-dim sphere, radius r,in E^5
Phi=Sph.embedding()                  # embedding S^4 -> E^5
E=Sph.ambient()                      # ambient space E^5
print("In",E)
print("the spherical coordinates are")
print(E.cartesian_coordinates()[:],"=")
Phi.coord_functions()[:]             # print coordinate functions

In 5-dimensional Euclidean space E^5
the spherical coordinates are
(x1, x2, x3, x4, x5) =

(r*cos(phi_4)*sin(phi_1)*sin(phi_2)*sin(phi_3),
 r*sin(phi_1)*sin(phi_2)*sin(phi_3)*sin(phi_4),
 r*cos(phi_3)*sin(phi_1)*sin(phi_2),
 r*cos(phi_2)*sin(phi_1),
 r*cos(phi_1))

Using the result we can define the transition spherU -> cartU in $R^5$.

R5=Manifold(5,'R^5')                      # manifold R^5
cart.<x1,x2,x3,x4,x5>=R5.chart()          # Cartesian coord. in R^5
U=R5.open_subset('U',                     # open subset of R^5,
    coord_def={cart:(x3!=0,x2!=0,x1<0)})  #{x3=0,x2=0,x1>=0} exclud.
cartU=cart.restrict(U)                    # restrict Cart.coord.to U
# use the same names as in the output of the previous command
spherU.<r,phi_1,phi_2,phi_3,phi_4>=U.chart(r'r:(0,+oo) phi1:(0,pi):\phi_1\
phi2:(0,pi):\phi_2 phi3:0,pi):\phi_3 phi4:(-pi,pi):periodic:\phi_4')
                                          # spherical coordinates on U
spher_to_cart = spherU.transition_map(cartU,     # define transition
(r*cos(phi_4)*sin(phi_1)*sin(phi_2)*sin(phi_3),  # spher -> cart
r*sin(phi_1)*sin(phi_2)*sin(phi_3)*sin(phi_4),
r*cos(phi_3)*sin(phi_1)*sin(phi_2),
r*cos(phi_2)*sin(phi_1),
r*cos(phi_1)))

spher_to_cart.disp()                      # show transition 

x1 = r*cos(phi4)*sin(phi1)*sin(phi2)*sin(phi3)
x2 = r*sin(phi1)*sin(phi2)*sin(phi3)*sin(phi4)
x3 = r*cos(phi3)*sin(phi1)*sin(phi2)
x4 = r*cos(phi2)*sin(phi1)
x5 = r*cos(phi1)

Inverse transition:

fun=(sqrt(x1^2+x2^2+x3^2+x4^2+x5^2),        # functions defining the 
     atan2(sqrt(x1^2+x2^2+x3^2+x4^2),x5),   # inverse transition
     atan2(sqrt(x1^2+x2^2+x3^2),x4),
     atan2(sqrt(x1^2+x2^2),x3),
     atan2(x2,x1))
cart_to_spher=spher_to_cart.set_inverse(*fun)  # define inv. trans.

Check of the inverse coordinate transformation:
  r == r  *passed*
  phi1 == arctan2(r*sin(phi1), r*cos(phi1))  **failed**
  phi2 == arctan2(r*sin(phi1)*sin(phi2), r*cos(phi2)*sin(phi1))  **failed**
  phi3 == arctan2(r*abs(sin(phi3))*sin(phi1)*sin(phi2), r*cos(phi3)*sin(phi1)*sin(phi2))  **failed**
  phi4 == arctan2(r*sin(phi1)*sin(phi2)*sin(phi3)*sin(phi4), r*cos(phi4)*sin(phi1)*sin(phi2)*sin(phi3))  **failed**
  x1 == x1  *passed*
  x2 == x2  *passed*
  x3 == x3  *passed*
  x4 == x4  *passed*
  x5 == x5  *passed*
NB: a failed report can reflect a mere lack of simplification.

What's next?

Take a look at the notebook The notion of module.