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sagemanifolds
GitHub Repository: sagemanifolds/IntroToManifolds
Path: blob/main/05Manifold_Spher_Higher_Dim.ipynb
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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 n3n\leq 3 (SageMath 9.5).


Example 5.1

Let us demonstrate spherical coordinates for n=3n=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 E4E^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

ι:S3E4on A:(χ,θ,ϕ)(x1,x2,x3,x4)=(cos(ϕ)sin(χ)sin(θ),sin(χ)sin(ϕ)sin(θ),cos(θ)sin(χ),cos(χ))\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 S3S^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

ι:S3E4on A:(χ,θ,ϕ)(x1,x2,x3,x4)=(cos(ϕ)sin(χ)sin(θ),sin(χ)sin(ϕ)sin(θ),cos(θ)sin(χ),cos(χ))on S3{NP}:(u,v,w)(x1,x2,x3,x4)=(2uu2+v2+w2+1,2vu2+v2+w2+1,2wu2+v2+w2+1,u2+v2+w21u2+v2+w2+1)on S3{SP}:(u,v,w)(x1,x2,x3,x4)=(2uu2+v2+w2+1,2vu2+v2+w2+1,2wu2+v2+w2+1,u2+v2+w21u2+v2+w2+1)\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 S3S^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()

{u=uu2+v2+w2v=vu2+v2+w2w=wu2+v2+w2\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()

{u=uu2+v2+w2v=vu2+v2+w2w=wu2+v2+w2\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 R4R^4.

We can use manifolds.Sphere to obtain the transition from spherical to Cartesian coordinates in R4R^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 R4R^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 R5R^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 R5R^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.