{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Sphere $\\mathbb{S}^2$\n", "\n", "This notebook demonstrates some differential geometry capabilities of SageMath on the example of the 2-dimensional sphere. The corresponding tools have been developed within\n", "the [SageManifolds](http://sagemanifolds.obspm.fr) project.\n", "\n", "Click [here](https://raw.githubusercontent.com/sagemanifolds/SageManifolds/master/Notebooks/SM_sphere_S2.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 9.2 is required to run this notebook:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 9.2.beta12, Release Date: 2020-09-06'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we set up the notebook to display math formulas using LaTeX formatting:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and we initialize a time counter for benchmarking:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "import time\n", "comput_time0 = time.perf_counter()" ] }, { "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", "\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 $\\{0,1\\}$." ] }, { "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/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbb{S}^2\n", "\\end{math}" ], "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/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "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/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbf{Smooth}_{\\Bold{R}}\n", "\\end{math}" ], "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": [ "### 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": 9, "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": 10, "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": 11, "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": 12, "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": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(x, y)\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (U, (x, y))" ] }, "execution_count": 13, "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": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}x\n", "\\end{math}" ], "text/plain": [ "x" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN[1]" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 15, "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": 16, "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": 17, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(V,({x'}, {y'})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (V, (xp, yp))" ] }, "execution_count": 17, "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": 18, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(U,(x, y)\\right), \\left(V,({x'}, {y'})\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[Chart (U, (x, y)), Chart (V, (xp, yp))]" ] }, "execution_count": 18, "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": 19, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} {x'} & = & \\frac{x}{x^{2} + y^{2}} \\\\ {y'} & = & \\frac{y}{x^{2} + y^{2}} \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "xp = x/(x^2 + y^2)\n", "yp = y/(x^2 + y^2)" ] }, "execution_count": 19, "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+yp^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": 20, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} x & = & \\frac{{x'}}{{x'}^{2} + {y'}^{2}} \\\\ y & = & \\frac{{y'}}{{x'}^{2} + {y'}^{2}} \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "x = xp/(xp^2 + yp^2)\n", "y = yp/(xp^2 + yp^2)" ] }, "execution_count": 20, "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": 21, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(U,(x, y)\\right), \\left(V,({x'}, {y'})\\right), \\left(W,(x, y)\\right), \\left(W,({x'}, {y'})\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[Chart (U, (x, y)),\n", " Chart (V, (xp, yp)),\n", " Chart (W, (x, y)),\n", " Chart (W, (xp, yp))]" ] }, "execution_count": 21, "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": 22, "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": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(W,(x, y)\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (W, (x, y))" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN_W = stereoN.restrict(W)\n", "stereoN_W" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(W,({x'}, {y'})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (W, (xp, yp))" ] }, "execution_count": 24, "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": 25, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 18 graphics primitives" ] }, "execution_count": 25, "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": 26, "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": 27, "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": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(A,(x, y)\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (A, (x, y))" ] }, "execution_count": 28, "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": 29, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(A,({\\theta}, {\\phi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (A, (th, ph))" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher. = A.chart(r'th:(0,pi):\\theta ph:(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": 30, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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.\n", "\\end{math}" ], "text/plain": [ "x = -cos(ph)*sin(th)/(cos(th) - 1)\n", "y = -sin(ph)*sin(th)/(cos(th) - 1)" ] }, "execution_count": 30, "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:" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Check of the inverse coordinate transformation:\n", " th == 2*arctan(sqrt(-cos(th) + 1)/sqrt(cos(th) + 1)) **failed**\n", " ph == pi + arctan2(sin(ph)*sin(th)/(cos(th) - 1), cos(ph)*sin(th)/(cos(th) - 1)) **failed**\n", " x == x *passed*\n", " y == y *passed*\n", "NB: a failed report can reflect a mere lack of simplification.\n" ] } ], "source": [ "spher_to_stereoN.set_inverse(2*atan(1/sqrt(x^2+y^2)), atan2(-y,-x)+pi)" ] }, { "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": 32, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} {\\theta} & = & 2 \\, \\arctan\\left(\\frac{1}{\\sqrt{x^{2} + y^{2}}}\\right) \\\\ {\\phi} & = & \\pi + \\arctan\\left(-y, -x\\right) \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "th = 2*arctan(1/sqrt(x^2 + y^2))\n", "ph = pi + arctan2(-y, -x)" ] }, "execution_count": 32, "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": 33, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} {x'} & = & -\\frac{\\cos\\left({\\phi}\\right) \\cos\\left({\\theta}\\right) - \\cos\\left({\\phi}\\right)}{\\sin\\left({\\theta}\\right)} \\\\ {y'} & = & -\\frac{\\cos\\left({\\theta}\\right) \\sin\\left({\\phi}\\right) - \\sin\\left({\\phi}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "xp = -(cos(ph)*cos(th) - cos(ph))/sin(th)\n", "yp = -(cos(th)*sin(ph) - sin(ph))/sin(th)" ] }, "execution_count": 33, "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": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} {\\theta} & = & 2 \\, \\arctan\\left(\\sqrt{{x'}^{2} + {y'}^{2}}\\right) \\\\ {\\phi} & = & \\pi - \\arctan\\left(\\frac{{y'}}{{x'}^{2} + {y'}^{2}}, -\\frac{{x'}}{{x'}^{2} + {y'}^{2}}\\right) \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "th = 2*arctan(sqrt(xp^2 + yp^2))\n", "ph = pi - arctan2(yp/(xp^2 + yp^2), -xp/(xp^2 + yp^2))" ] }, "execution_count": 34, "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": 35, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(U,(x, y)\\right), \\left(V,({x'}, {y'})\\right), \\left(W,(x, y)\\right), \\left(W,({x'}, {y'})\\right), \\left(A,(x, y)\\right), \\left(A,({x'}, {y'})\\right), \\left(A,({\\theta}, {\\phi})\\right)\\right]\n", "\\end{math}" ], "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, (th, ph))]" ] }, "execution_count": 35, "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": 36, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 30 graphics primitives" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher.plot(stereoN, number_values=15, ranges={th: (pi/8,pi)})" ] }, { "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": 37, "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": 38, "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": 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((0,0), name='N') ; print(N)\n", "S = U((0,0), name='S') ; print(S)" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}V\n", "\\end{math}" ], "text/plain": [ "Open subset V of the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N.parent()" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}U\n", "\\end{math}" ], "text/plain": [ "Open subset U of the 2-dimensional differentiable manifold S^2" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S.parent()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We have of course

" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N in V" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N in S2" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{False}\n", "\\end{math}" ], "text/plain": [ "False" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N in U" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{False}\n", "\\end{math}" ], "text/plain": [ "False" ] }, "execution_count": 45, "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": 46, "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": 47, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 47, "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": 48, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\frac{1}{2} \\, \\pi, \\frac{1}{2} \\, \\pi\\right)\n", "\\end{math}" ], "text/plain": [ "(1/2*pi, 1/2*pi)" ] }, "execution_count": 48, "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": 49, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Error: the point does not belong to the domain of Chart (A, (th, ph))\n" ] } ], "source": [ "try:\n", " N.coord(spher)\n", "except ValueError as exc:\n", " print('Error: ' + str(exc))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Maps between manifolds: the embedding of $\\mathbb{S}^2$ into $\\mathbb{R}^3$\n", "\n", "Let us first declare $\\mathbb{R}^3$ as the 3-dimensional Euclidean space, denoting the Cartesian coordinates by\n", "$(X,Y,Z)$:" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{R}^3,(X, Y, Z)\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (R^3, (X, Y, Z))" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R3. = EuclideanSpace(name='R^3', latex_name=r'\\mathbb{R}^3', metric_name='h')\n", "cart = R3.cartesian_coordinates()\n", "cart" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The embedding of $\\mathbb{S}^2$ into $\\mathbb{R}^3$ is then defined by the standard formulas relating the stereographic coordinates to the ambient Cartesian ones when considering the **stereographic projection** from the point $(0,0,1)$ (North pole) or $(0, 0, -1)$ (South pole) to the equatorial plane $Z=0$:" ] }, { "cell_type": "code", "execution_count": 51, "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": 52, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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) \\end{array}\n", "\\end{math}" ], "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": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.display()" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Hom}\\left(\\mathbb{S}^2,\\mathbb{R}^3\\right)\n", "\\end{math}" ], "text/plain": [ "Set of Morphisms from 2-dimensional differentiable manifold S^2 to Euclidean space R^3 in Category of smooth manifolds over Real Field with 53 bits of precision" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.parent()" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Set of Morphisms from 2-dimensional differentiable manifold S^2 to Euclidean space 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": 55, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 55, "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": 56, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point Phi(N) on the Euclidean space R^3\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0, 0, 1\\right)\n", "\\end{math}" ], "text/plain": [ "(0, 0, 1)" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N1 = Phi(N) ; print(N1) ; N1 ; N1.coord()" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point Phi(S) on the Euclidean space R^3\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0, 0, -1\\right)\n", "\\end{math}" ], "text/plain": [ "(0, 0, -1)" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S1 = Phi(S) ; print(S1) ; S1 ; S1.coord()" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point Phi(E) on the Euclidean space R^3\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0, 1, 0\\right)\n", "\\end{math}" ], "text/plain": [ "(0, 1, 0)" ] }, "execution_count": 58, "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": 59, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\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)\n", "\\end{math}" ], "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": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.expr(stereoN_A, cart)" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), \\cos\\left({\\theta}\\right)\\right)\n", "\\end{math}" ], "text/plain": [ "(cos(ph)*sin(th), sin(ph)*sin(th), cos(th))" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.expr(spher, cart)" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\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}\n", "\\end{math}" ], "text/plain": [ "Phi: S^2 --> R^3\n", "on A: (th, ph) |--> (X, Y, Z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th))" ] }, "execution_count": 61, "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": 62, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph_spher = spher.plot(chart=cart, mapping=Phi, number_values=11, \n", " color='blue', label_axes=False)\n", "graph_spher" ] }, { "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": 63, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph_stereoN = stereoN.plot(chart=cart, mapping=Phi, number_values=25, \n", " label_axes=False)\n", "graph_stereoN" ] }, { "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": 64, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "