{ "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 9.3.beta2, Release Date: 2020-11-24'" ] }, "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", "