{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Sphere $\\mathbb{S}^2$ (SymPy version)\n", "\n", "This notebook demonstrates a few differential geometry capabilities of [SageMath](http://www.sagemath.org/) on the example of the 2-dimensional sphere, using [SymPy](http://www.sympy.org) as symbolic engine, instead of SageMath's default one (Pynac+Maxima). The relevent tools have been implemented through the \n", "[SageManifolds](http://sagemanifolds.obspm.fr) project.\n", "\n", "Click [here](https://raw.githubusercontent.com/sagemanifolds/SageManifolds/master/Notebooks/SM_sphere_S2_sympy.ipynb) to download the notebook file (ipynb format). To run it, you must start SageMath with the Jupyter interface, via the command `sage -n jupyter`" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*NB:* a version of SageMath at least equal to 8.3 is required to run this worksheet:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 8.8, Release Date: 2019-06-26'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we set up the notebook to display mathematical objects using LaTeX rendering:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%display latex\n", "from sympy import init_printing\n", "init_printing()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We also define a viewer for 3D plots (use `'threejs'` or `'jmol'` for interactive 3D graphics):" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "viewer3D = 'threejs' # must be 'threejs', jmol', 'tachyon' or None (default)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## $\\mathbb{S}^2$ as a 2-dimensional differentiable manifold\n", "\n", "We start by declaring $\\mathbb{S}^2$ as a differentiable manifold of dimension 2 over $\\mathbb{R}$:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "S2 = Manifold(2, 'S^2', latex_name=r'\\mathbb{S}^2', start_index=1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The first argument, 2, is the dimension of the manifold, while the second argument is the symbol used to label the manifold.

\n", "

The argument start_index sets the index range to be used on the manifold for labelling components w.r.t. a basis or a frame: start_index=1 corresponds to $\\{1,2\\}$; the default value is start_index=0 and yields to $\\{0,1\\}$.

" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "print(S2)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "2-dimensional differentiable manifold S^2" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The manifold is a `Parent` object:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "True" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "isinstance(S2, Parent)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

in the category of smooth manifolds over $\\mathbb{R}$:

" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Category of smooth manifolds over Real Field with 53 bits of precision" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.category()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We ask for all symbolic computations to be performed with [SymPy](http://www.sympy.org), instead of SageMath's default symbolic engine (Pynac+Maxima, implemented via SageMath's Symbolic Ring):" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "S2.set_calculus_method('sympy')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Coordinate charts on $\\mathbb{S}^2$\n", "\n", "The sphere cannot be covered by a single chart. At least two charts are necessary, for instance the charts associated with the stereographic projections from the North pole and the South pole respectively. Let us introduce the open subsets covered by these two charts: \n", "$$ U := \\mathbb{S}^2\\setminus\\{N\\}, $$  \n", "$$ V := \\mathbb{S}^2\\setminus\\{S\\}, $$\n", "where $N$ is a point of $\\mathbb{S}^2$, which we shall call the North pole, and $S$ is the point of $U$ of stereographic coordinates $(0,0)$, which we call the South pole:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Open subset U of the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "U = S2.open_subset('U') ; print(U)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Open subset V of the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "V = S2.open_subset('V') ; print(V)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We declare that $\\mathbb{S}^2 = U \\cup V$:

" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "S2.declare_union(U, V)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Then we declare the stereographic chart on $U$, denoting by $(x,y)$ the coordinates resulting from the stereographic projection from the North pole:

" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [], "source": [ "stereoN. = U.chart()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The expression `.` in the left-hand side means that the Python variables `x` and `y` are set to the two coordinates of the chart. This allows one to refer subsequently to the coordinates by their names. In the present case, the function `chart()` has no argument, which implies that the coordinate symbols will be `x` and `y` (i.e. exactly the characters appearing in the `<...>` operator) and that each coordinate range is $(-\\infty,+\\infty)$. As we will see below, for other cases, an argument must be passed to `chart()` to specify each coordinate symbol and range, as well as some specific LaTeX symbol." ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (U, (x, y))" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The coordinates can be accessed individually, either by means of their indices in the chart ( following the convention `start_index=1` set in the manifold's definition) or by their names as Python variables:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "x" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN[1]" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "True" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y is stereoN[2]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Similarly, we introduce on $V$ the coordinates $(x',y')$ corresponding to the stereographic projection from the South pole:

" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [], "source": [ "stereoS. = V.chart(\"xp:x' yp:y'\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this case, the string argument passed to `chart` stipulates that the text-only names of the coordinates are xp and yp (same as the Python variables names defined within the `<...>` operator in the left-hand side), while their LaTeX names are $x'$ and $y'$." ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (V, (xp, yp))" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

At this stage, the user's atlas on the manifold has two charts:

" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[Chart (U, (x, y)), Chart (V, (xp, yp))]" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We have to specify the transition map between the charts 'stereoN' = $(U,(x,y))$ and 'stereoS' = $(V,(x',y'))$; it is given by the standard inversion formulas:

" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "xp = x/(x^2 + y^2)\n", "yp = y/(x^2 + y^2)" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN_to_S = stereoN.transition_map(stereoS, \n", " (x/(x^2+y^2), y/(x^2+y^2)), \n", " intersection_name='W',\n", " restrictions1= x^2+y^2!=0, \n", " restrictions2= xp^2+xp^2!=0)\n", "stereoN_to_S.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the above declaration, 'W' is the name given to the chart-overlap subset: $W := U\\cap V$, the condition $x^2+y^2 \\not=0$  defines $W$ as a subset of $U$, and the condition $x'^2+y'^2\\not=0$ defines $W$ as a subset of $V$.\n", "\n", "The inverse coordinate transformation is computed by means of the method `inverse()`:" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "x = xp/(xp^2 + yp^2)\n", "y = yp/(xp^2 + yp^2)" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS_to_N = stereoN_to_S.inverse()\n", "stereoS_to_N.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

In the present case, the situation is of course perfectly symmetric regarding the coordinates $(x,y)$ and $(x',y')$.

\n", "

At this stage, the user's atlas has four charts:

" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[Chart (U, (x, y)),\n", " Chart (V, (xp, yp)),\n", " Chart (W, (x, y)),\n", " Chart (W, (xp, yp))]" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Let us store $W = U\\cap V$ into a Python variable for future use:

" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [], "source": [ "W = U.intersection(V)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Similarly we store the charts $(W,(x,y))$ (the restriction of  $(U,(x,y))$ to $W$) and $(W,(x',y'))$ (the restriction of $(V,(x',y'))$ to $W$) into Python variables:

" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (W, (x, y))" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN_W = stereoN.restrict(W)\n", "stereoN_W" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (W, (xp, yp))" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS_W = stereoS.restrict(W)\n", "stereoS_W" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We may plot the chart $(W, (x',y'))$ in terms of itself, as a grid:

" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 17 graphics primitives" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS_W.plot()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

More interestingly, let us plot the stereographic chart $(x',y')$ in terms of the stereographic chart $(x,y)$ on the domain $W$ where both systems overlap (we split the plot in four parts to avoid the singularity at $(x',y')=(0,0)$):

" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 72 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graphSN1 = stereoS_W.plot(stereoN, ranges={xp:[-6,-0.02], yp:[-6,-0.02]})\n", "graphSN2 = stereoS_W.plot(stereoN, ranges={xp:[-6,-0.02], yp:[0.02,6]})\n", "graphSN3 = stereoS_W.plot(stereoN, ranges={xp:[0.02,6], yp:[-6,-0.02]})\n", "graphSN4 = stereoS_W.plot(stereoN, ranges={xp:[0.02,6], yp:[0.02,6]})\n", "show(graphSN1+graphSN2+graphSN3+graphSN4,\n", " xmin=-1.5, xmax=1.5, ymin=-1.5, ymax=1.5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Spherical coordinates

\n", "

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:

" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Open subset A of the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "A = W.open_subset('A', coord_def={stereoN_W: (y!=0, x<0), \n", " stereoS_W: (yp!=0, xp<0)})\n", "print(A)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The restriction of the stereographic chart from the North pole to $A$ is

" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (A, (x, y))" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN_A = stereoN_W.restrict(A)\n", "stereoN_A" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We then declare the chart $(A,(\\theta,\\phi))$ by specifying the intervals $(0,\\pi)$ and $(0,2\\pi)$ spanned by respectively $\\theta$ and $\\phi$:

" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (A, (theta, phi))" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher. = A.chart(r'theta:(0,pi):\\theta phi:(0,2*pi):\\phi') ; spher" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The specification of the spherical coordinates is completed by providing the transition map with the stereographic chart $(A,(x,y))$:

" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "x = -cos(phi)*sin(theta)/(cos(theta) - 1)\n", "y = -sin(phi)*sin(theta)/(cos(theta) - 1)" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher_to_stereoN = spher.transition_map(stereoN_A, \n", " (sin(th)*cos(ph)/(1-cos(th)),\n", " sin(th)*sin(ph)/(1-cos(th))))\n", "spher_to_stereoN.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We also provide the inverse transition map, asking to check that the provided formulas are indeed correct (argument `verbose=True`):" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Check of the inverse coordinate transformation:\n", " theta == 2*arctan(sqrt(-cos(theta) + 1)/sqrt(cos(theta) + 1))\n", " phi == pi + arctan2(sin(phi)*sin(theta)/(cos(theta) - 1), cos(phi)*sin(theta)/(cos(theta) - 1))\n", " x == x\n", " y == y\n" ] } ], "source": [ "spher_to_stereoN.set_inverse(2*atan(1/sqrt(x^2+y^2)), atan2(-y,-x)+pi,\n", " verbose=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The check is passed, modulo some lack of trigonometric simplifications in the first two lines." ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "theta = 2*arctan(1/sqrt(x^2 + y^2))\n", "phi = pi + arctan2(-y, -x)" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher_to_stereoN.inverse().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The transition map $(A,(\\theta,\\phi))\\rightarrow (A,(x',y'))$ is obtained by combining the transition maps $(A,(\\theta,\\phi))\\rightarrow (A,(x,y))$ and $(A,(x,y))\\rightarrow (A,(x',y'))$:" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "xp = -(cos(phi)*cos(theta) - cos(phi))/sin(theta)\n", "yp = -(cos(theta)*sin(phi) - sin(phi))/sin(theta)" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN_to_S_A = stereoN_to_S.restrict(A)\n", "spher_to_stereoS = stereoN_to_S_A * spher_to_stereoN\n", "spher_to_stereoS.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Similarly, the transition map $(A,(x',y'))\\rightarrow (A,(\\theta,\\phi))$ is obtained by combining the transition maps $(A,(x',y'))\\rightarrow (A,(x,y))$ and $(A,(x,y))\\rightarrow (A,(\\theta,\\phi))$:" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "theta = 2*arctan(sqrt(xp^2 + yp^2))\n", "phi = pi - arctan2(yp/(xp^2 + yp^2), -xp/(xp^2 + yp^2))" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS_to_N_A = stereoN_to_S.inverse().restrict(A)\n", "stereoS_to_spher = spher_to_stereoN.inverse() * stereoS_to_N_A \n", "stereoS_to_spher.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The user atlas of $\\mathbb{S}^2$ is now

" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[Chart (U, (x, y)),\n", " Chart (V, (xp, yp)),\n", " Chart (W, (x, y)),\n", " Chart (W, (xp, yp)),\n", " Chart (A, (x, y)),\n", " Chart (A, (xp, yp)),\n", " Chart (A, (theta, phi))]" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Let us draw the grid of spherical coordinates $(\\theta,\\phi)$ in terms of stereographic coordinates from the North pole $(x,y)$:

" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 30 graphics primitives" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher.plot(stereoN, number_values=15, ranges={th: (pi/8,pi)})" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Conversly, we may represent the grid of the stereographic coordinates $(x,y)$ restricted to $A$ in terms of the spherical coordinates $(\\theta,\\phi)$. We limit ourselves to one quarter (cf. the argument ranges):

" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 40 graphics primitives" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN_A.plot(spher, ranges={x: (0.01,8), y: (0.01,8)}, number_values=20, plot_points=200)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Points on $\\mathbb{S}^2$

\n", "

We declare the North pole (resp. the South pole) as the point of coordinates $(0,0)$ in the chart $(V,(x',y'))$ (resp. in the chart $(U,(x,y))$):

" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point N on the 2-dimensional differentiable manifold S^2\n", "Point S on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "N = V.point((0,0), chart=stereoS, name='N') ; print(N)\n", "S = U.point((0,0), chart=stereoN, name='S') ; print(S)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Since points are Sage Element's, the corresponding Parent being the manifold subsets, an equivalent writing of the above declarations is

" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point N on the 2-dimensional differentiable manifold S^2\n", "Point S on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "N = V((0,0), chart=stereoS, name='N') ; print(N)\n", "S = U((0,0), chart=stereoN, name='S') ; print(S)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Moreover, since stereoS in the default chart on $V$ and stereoN is the default one on $U$, their mentions can be omitted, so that the above can be shortened to

" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point N on the 2-dimensional differentiable manifold S^2\n", "Point S on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "N = V((0,0), name='N') ; print(N)\n", "S = U((0,0), name='S') ; print(S)" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Open subset V of the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N.parent()" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Open subset U of the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S.parent()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We have of course

" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "True" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N in V" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "True" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N in S2" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "False" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N in U" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "False" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N in A" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Let us introduce some point at the equator:

" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [], "source": [ "E = S2((0,1), chart=stereoN, name='E')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The point $E$ is in the open subset $A$:

" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "True" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E in A" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We may then ask for its spherical coordinates $(\\theta,\\phi)$:

" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "(1/2*pi, 1/2*pi)" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E.coord(spher)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

which is not possible for the point $N$:

" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Error: the point does not belong to the domain of Chart (A, (theta, phi))\n" ] } ], "source": [ "try:\n", " N.coord(spher)\n", "except ValueError as exc:\n", " print('Error: ' + str(exc))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Mappings between manifolds: the embedding of $\\mathbb{S}^2$ into $\\mathbb{R}^3$

\n", "

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

" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R^3, (X, Y, Z))" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R3 = Manifold(3, 'R^3', r'\\mathbb{R}^3', start_index=1)\n", "R3.set_calculus_method('sympy')\n", "cart. = R3.chart() ; cart" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [], "source": [ "Phi = S2.diff_map(R3, {(stereoN, cart): \n", " [2*x/(1+x^2+y^2), 2*y/(1+x^2+y^2),\n", " (x^2+y^2-1)/(1+x^2+y^2)],\n", " (stereoS, cart): \n", " [2*xp/(1+xp^2+yp^2), 2*yp/(1+xp^2+yp^2),\n", " (1-xp^2-yp^2)/(1+xp^2+yp^2)]},\n", " name='Phi', latex_name=r'\\Phi')" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Phi: S^2 --> R^3\n", "on U: (x, y) |--> (X, Y, Z) = (2*x/(x^2 + y^2 + 1), 2*y/(x^2 + y^2 + 1), (x^2 + y^2 - 1)/(x^2 + y^2 + 1))\n", "on V: (xp, yp) |--> (X, Y, Z) = (2*xp/(xp^2 + yp^2 + 1), 2*yp/(xp^2 + yp^2 + 1), -(xp^2 + yp^2 - 1)/(xp^2 + yp^2 + 1))" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.display()" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Set of Morphisms from 2-dimensional differentiable manifold S^2 to 3-dimensional differentiable manifold R^3 in Category of smooth manifolds over Real Field with 53 bits of precision" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.parent()" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Set of Morphisms from 2-dimensional differentiable manifold S^2 to 3-dimensional differentiable manifold R^3 in Category of smooth manifolds over Real Field with 53 bits of precision\n" ] } ], "source": [ "print(Phi.parent())" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "True" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.parent() is Hom(S2, R3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

$\\Phi$ maps points of $\\mathbb{S}^2$ to points of $\\mathbb{R}^3$:

" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point Phi(N) on the 3-dimensional differentiable manifold R^3\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "(0, 0, 1)" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N1 = Phi(N) ; print(N1) ; N1 ; N1.coord()" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point Phi(S) on the 3-dimensional differentiable manifold R^3\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "(0, 0, -1)" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S1 = Phi(S) ; print(S1) ; S1 ; S1.coord()" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point Phi(E) on the 3-dimensional differentiable manifold R^3\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "(0, 1, 0)" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E1 = Phi(E) ; print(E1) ; E1 ; E1.coord()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

$\\Phi$ has been defined in terms of the stereographic charts $(U,(x,y))$ and $(V,(x',y'))$, but we may ask its expression in terms of spherical coordinates. The latter is then computed by means of the transition map $(A,(x,y))\\rightarrow (A,(\\theta,\\phi))$:

" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "(2*x/(x^2 + y^2 + 1), 2*y/(x^2 + y^2 + 1), (x^2 + y^2 - 1)/(x^2 + y^2 + 1))" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.expr(stereoN_A, cart)" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "(cos(phi)*sin(theta), sin(phi)*sin(theta), cos(theta))" ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.expr(spher, cart)" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Phi: S^2 --> R^3\n", "on A: (theta, phi) |--> (X, Y, Z) = (cos(phi)*sin(theta), sin(phi)*sin(theta), cos(theta))" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.display(spher, cart)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph_spher = spher.plot(chart=cart, mapping=Phi, number_values=11, \n", " color='blue', label_axes=False)\n", "show(graph_spher, viewer=viewer3D)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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:

" ] }, { "cell_type": "code", "execution_count": 65, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph_stereoN = stereoN.plot(chart=cart, mapping=Phi, number_values=25, \n", " label_axes=False)\n", "show(graph_stereoN, viewer=viewer3D)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "