3D Plots of charts, vector fields and curves on $\mathbb{S}^2$

This worksheet is based on SageMath 7.5.beta4 and the three.js viewer developed at trac #12402.

NB: a version of SageMath at least equal to 7.5.beta4 is required:

version()

'SageMath version 7.5.beta4, Release Date: 2016-11-24'

First we set up the notebook to display mathematical objects using LaTeX formatting:

%display latex

We also define a viewer for 3D plots (use 'threejs' or 'jmol' for interactive 3D graphics):

viewer3D = 'threejs' # must be 'jmol', 'tachyon', 'threejs' or None (default)

$\mathbb{S}^2$ as a 2-dimensional differentiable manifold

We declare $\mathbb{S}^2$ as a differentiable manifold of dimension 2 over $\mathbb{R}$ and we endow it with the stereographic charts from the North pole (stereoN) and the South pole (stereoS):

S2 = Manifold(2, 'S^2', latex_name=r'\mathbb{S}^2', start_index=1)
U = S2.open_subset('U')
V = S2.open_subset('V')
S2.declare_union(U, V)
stereoN.<x,y> = U.chart()
stereoS.<xp,yp> = V.chart(r"xp:x' yp:y'")
stereoN_to_S = stereoN.transition_map(stereoS, (x/(x^2+y^2), y/(x^2+y^2)), 
                                      intersection_name='W',
                                      restrictions1= x^2+y^2!=0, 
                                      restrictions2= xp^2+xp^2!=0)
stereoS_to_N = stereoN_to_S.inverse()
W = U.intersection(V)
eU = stereoN.frame()
eV = stereoS.frame()
S2.frames()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(U, \left(\frac{\partial}{\partial x },\frac{\partial}{\partial y }\right)\right), \left(V, \left(\frac{\partial}{\partial {x'} },\frac{\partial}{\partial {y'} }\right)\right), \left(W, \left(\frac{\partial}{\partial x },\frac{\partial}{\partial y }\right)\right), \left(W, \left(\frac{\partial}{\partial {x'} },\frac{\partial}{\partial {y'} }\right)\right)\right]$$

Spherical coordinates

The standard spherical (or polar) coordinates $(\theta,\phi)$ are defined on the open domain $A\subset W \subset \mathbb{S}^2$ that is the complement of the "origin meridian"; since the latter is the half-circle defined by $y=0$ and $x\geq 0$, we declare:

A = W.open_subset('A', coord_def={stereoN.restrict(W): (y!=0, x<0), 
                                  stereoS.restrict(W): (yp!=0, xp<0)})
spher.<th,ph> = A.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
spher_to_stereoN = spher.transition_map(stereoN.restrict(A), 
                                        (sin(th)*cos(ph)/(1-cos(th)),
                                         sin(th)*sin(ph)/(1-cos(th))) )
spher_to_stereoN.set_inverse(2*atan(1/sqrt(x^2+y^2)), atan2(-y,-x)+pi)
spher_to_stereoN.display()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left\{\begin{array}{lcl} x & = & -\frac{\cos\left({\phi}\right) \sin\left({\theta}\right)}{\cos\left({\theta}\right) - 1} \\ y & = & -\frac{\sin\left({\phi}\right) \sin\left({\theta}\right)}{\cos\left({\theta}\right) - 1} \end{array}\right.$$

Embedding of $\mathbb{S}^2$ into $\mathbb{R}^3$

Let us first declare $\mathbb{R}^3$ as a 3-dimensional manifold covered by a single chart (Cartesian coordinates):

R3 = Manifold(3, 'R^3', r'\mathbb{R}^3', start_index=1)
cart.<X,Y,Z> = R3.chart() ; cart

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\mathbb{R}^3,(X, Y, Z)\right)$$

The embedding of the sphere is defined as a differential mapping $\Phi: \mathbb{S}^2 \rightarrow \mathbb{R}^3$:

Phi = S2.diff_map(R3, {(stereoN, cart): [2*x/(1+x^2+y^2), 2*y/(1+x^2+y^2),
                                             (x^2+y^2-1)/(1+x^2+y^2)],
                       (stereoS, cart): [2*xp/(1+xp^2+yp^2), 2*yp/(1+xp^2+yp^2),
                                             (1-xp^2-yp^2)/(1+xp^2+yp^2)]},
                  name='Phi', latex_name=r'\Phi')

Phi.expr(stereoN.restrict(A), cart)
Phi.display()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} \Phi:& \mathbb{S}^2 & \longrightarrow & \mathbb{R}^3 \\ \mbox{on}\ U : & \left(x, y\right) & \longmapsto & \left(X, Y, Z\right) = \left(\frac{2 \, x}{x^{2} + y^{2} + 1}, \frac{2 \, y}{x^{2} + y^{2} + 1}, \frac{x^{2} + y^{2} - 1}{x^{2} + y^{2} + 1}\right) \\ \mbox{on}\ V : & \left({x'}, {y'}\right) & \longmapsto & \left(X, Y, Z\right) = \left(\frac{2 \, {x'}}{{x'}^{2} + {y'}^{2} + 1}, \frac{2 \, {y'}}{{x'}^{2} + {y'}^{2} + 1}, -\frac{{x'}^{2} + {y'}^{2} - 1}{{x'}^{2} + {y'}^{2} + 1}\right) \\ \mbox{on}\ A : & \left({\theta}, {\phi}\right) & \longmapsto & \left(X, Y, Z\right) = \left(\cos\left({\phi}\right) \sin\left({\theta}\right), \sin\left({\phi}\right) \sin\left({\theta}\right), \cos\left({\theta}\right)\right) \end{array}$$

3D plots of coordinate charts on $\mathbb{S}^2$

Let us use $\Phi$ to draw the grid of spherical coordinates $(\theta,\phi)$ in terms of the Cartesian coordinates $(X,Y,Z)$ of $\mathbb{R}^3$:

graph_spher = spher.plot(chart=cart, mapping=Phi, number_values=11, color='blue',
                         label_axes=False)

show(graph_spher, viewer=viewer3D)

We may also use the embedding $\Phi$ to display the stereographic coordinate grid in terms of the Cartesian coordinates in $\mathbb{R}^3$. First for the stereographic coordinates from the North pole:

graph_stereoN = stereoN.plot(chart=cart, mapping=Phi, number_values=25,
                             label_axes=False)
show(graph_stereoN, viewer=viewer3D)

and then have a view with the stereographic coordinates from the South pole superposed (in green):

graph_stereoS = stereoS.plot(chart=cart, mapping=Phi, number_values=25, 
                             color='green', label_axes=False)
show(graph_stereoN + graph_stereoS, viewer=viewer3D)

Let us add the North and South poles to the graphic:

N = V.point((0,0), chart=stereoS, name='N')
S = U.point((0,0), chart=stereoN, name='S')
graph_N = N.plot(chart=cart, mapping=Phi, color='red', label_offset=0.05)
graph_S = S.plot(chart=cart, mapping=Phi, color='green', label_offset=0.05)
show(graph_stereoN + graph_stereoS + graph_N + graph_S, viewer=viewer3D)