{ "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](https://sagemanifolds.obspm.fr) project." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*NB:* a version of SageMath at least equal to 9.3 is required to run this notebook:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 10.1, Release Date: 2023-08-20'" ] }, "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": [ "## $\\mathbb{S}^2$ from the manifold catalog" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The 2-sphere, with predefined charts and embedding in the Euclidean 3-space, can be obtained directly from SageMath's manifold catalog:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathbb{S}^{2}\$" ], "text/latex": [ "$\\displaystyle \\mathbb{S}^{2}$" ], "text/plain": [ "2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2 = manifolds.Sphere(2)\n", "S2" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3\n" ] } ], "source": [ "print(S2)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(A,(\\theta, \\phi)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(A,(\\theta, \\phi)\\right)$" ], "text/plain": [ "Chart (A, (theta, phi))" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.spherical_coordinates()" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g = \\mathrm{d} \\theta\\otimes \\mathrm{d} \\theta + \\sin\\left(\\theta\\right)^{2} \\mathrm{d} \\phi\\otimes \\mathrm{d} \\phi\$" ], "text/latex": [ "$\\displaystyle g = \\mathrm{d} \\theta\\otimes \\mathrm{d} \\theta + \\sin\\left(\\theta\\right)^{2} \\mathrm{d} \\phi\\otimes \\mathrm{d} \\phi$" ], "text/plain": [ "g = dtheta⊗dtheta + sin(theta)^2 dphi⊗dphi" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.metric().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## $\\mathbb{S}^2$ defined from scratch as a 2-dimensional smooth manifold" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For the purpose of introducing generic smooth manifolds in SageMath, we shall not use the above predefined object. Instead we shall construct $\\mathbb{S}^2$ from scratch, by invoking the generic function `Manifold`:" ] }, { "cell_type": "code", "execution_count": 7, "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": "markdown", "metadata": {}, "source": [ "The function `Manifold` has actually many options, which are displayed via the command `Manifold?`:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "# Manifold?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "By default `Manifold` constructs a smooth manifold over $\\mathbb{R}$:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "print(S2)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathbb{S}^2\$" ], "text/latex": [ "$\\displaystyle \\mathbb{S}^2$" ], "text/plain": [ "2-dimensional differentiable manifold S^2" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\mathbb{S}^2$ is in the category of smooth manifolds over $\\mathbb{R}$:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbf{Smooth}_{\\Bold{R}}\$" ], "text/latex": [ "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbf{Smooth}_{\\Bold{R}}$" ], "text/plain": [ "Category of smooth manifolds over Real Field with 53 bits of precision" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.category()" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Category of smooth manifolds over Real Field with 53 bits of precision\n" ] } ], "source": [ "print(S2.category())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "At the moment, the real field $\\mathbb{R}$ is modeled by 53-bit floating-point approximations, but this plays no role in the manifold implementation:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Real Field with 53 bits of precision\n" ] } ], "source": [ "print(S2.base_field())" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.base_field() is RR" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Coordinate charts on $\\mathbb{S}^2$\n", "\n", "The function `Manifold` generates a manifold with no-predefined coordinate chart, so that the manifold (user) **atlas** is empty:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\right]$" ], "text/plain": [ "[]" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us introduce some charts. At least two charts are necessary to cover the sphere. Let us choose the charts associated with the **stereographic projections** to the equatorial plane from the North pole and the South pole respectively. We first 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": "markdown", "metadata": {}, "source": [ "To find the method to create an open subset, we type `U = S2.` to get the list of possible methods by autocompletion:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [], "source": [ "#U = S2." ] }, { "cell_type": "code", "execution_count": 17, "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')\n", "print(U)" ] }, { "cell_type": "code", "execution_count": 18, "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')\n", "print(V)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As an open subset of a smooth manifold, $U$ is itself a smooth manifold:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Join of Category of subobjects of sets and Category of smooth manifolds over Real Field with 53 bits of precision\n" ] } ], "source": [ "print(U.category())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [], "source": [ "S2.declare_union(U, V)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **stereographic chart** on $U$ is constructed from the stereographic projection from the North pole onto the equatorial plane: in the [Wikipedia figure](https://en.wikipedia.org/wiki/Stereographic_projection) below, the stereographic coordinates $(x,y)$ of the point $P\\in U$ are the Cartesian coordinates of the point $P'$ in the equatorial plane.\n", "\n", "![stereographic projection](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Stereoprojzero.svg/241px-Stereoprojzero.svg.png)\n", "\n", "We call this chart `stereoN` and construct it via the method `chart`:" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [], "source": [ "stereoN. = U.chart()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The syntax `.` in the left-hand side implies that the Python names `x` and `y` are added to the global namespace, to access to the two coordinates of the chart as symbolic variables. This allows one to refer subsequently to the coordinates by these names. Besides, 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": "markdown", "metadata": {}, "source": [ "*Note:* the notation `.` is not standard Python syntax, but a \"SageMath enhanced\" syntax. \n", "Actually the SageMath kernel preparses the cell entries before sending them to the Python interpreter. The outcome of the preparser is shown by the function `preparse`. In the present case:" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "stereoN = U.chart(names=('x', 'y',)); (x, y,) = stereoN._first_ngens(2)\n" ] } ], "source": [ "print(preparse(\"stereoN. = U.chart()\"))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Another example of preparsing:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\verb|Integer(2)**Integer(3)|\$" ], "text/latex": [ "$\\displaystyle \\verb|Integer(2)**Integer(3)|$" ], "text/plain": [ "'Integer(2)**Integer(3)'" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "preparse(\"2^3\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The chart created by the above command:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(U,(x, y)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(U,(x, y)\\right)$" ], "text/plain": [ "Chart (U, (x, y))" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (U, (x, y))\n" ] } ], "source": [ "print(stereoN)" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle x :\\ \\left( -\\infty, +\\infty \\right) ;\\quad y :\\ \\left( -\\infty, +\\infty \\right)\$" ], "text/latex": [ "$\\displaystyle x :\\ \\left( -\\infty, +\\infty \\right) ;\\quad y :\\ \\left( -\\infty, +\\infty \\right)$" ], "text/plain": [ "x: (-oo, +oo); y: (-oo, +oo)" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN.coord_range()" ] }, { "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": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle x\$" ], "text/latex": [ "$\\displaystyle x$" ], "text/plain": [ "x" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN[1]" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y is stereoN[2]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The coordinates are SageMath symbolic expressions:" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\verb|<class|\\verb| |\\verb|'sage.symbolic.expression.Expression'>|\$" ], "text/latex": [ "$\\displaystyle \\verb||$" ], "text/plain": [ "" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "type(y)" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\text{SR}\$" ], "text/latex": [ "$\\displaystyle \\text{SR}$" ], "text/plain": [ "Symbolic Ring" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y.parent()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Stereographic coordinates from the South Pole\n", "\n", "We introduce on $V$ the coordinates $(x',y')$ corresponding to the stereographic projection from the South pole:" ] }, { "cell_type": "code", "execution_count": 31, "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": 32, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(V,({x'}, {y'})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(V,({x'}, {y'})\\right)$" ], "text/plain": [ "Chart (V, (xp, yp))" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "At this stage, the user's atlas on the manifold is made of two charts:" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left(U,(x, y)\\right), \\left(V,({x'}, {y'})\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left(U,(x, y)\\right), \\left(V,({x'}, {y'})\\right)\\right]$" ], "text/plain": [ "[Chart (U, (x, y)), Chart (V, (xp, yp))]" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S2.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To complete the construction of the manifold structure, we have \n", "to specify the transition map between the charts `stereoN` = $(U,(x,y))$ and `stereoS` = $(V,(x',y'))$; it is given by standard inversion formulas:" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} {x'} & = & \\frac{x}{x^{2} + y^{2}} \\\\ {y'} & = & \\frac{y}{x^{2} + y^{2}} \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} {x'} & = & \\frac{x}{x^{2} + y^{2}} \\\\ {y'} & = & \\frac{y}{x^{2} + y^{2}} \\end{array}\\right.$" ], "text/plain": [ "xp = x/(x^2 + y^2)\n", "yp = y/(x^2 + y^2)" ] }, "execution_count": 34, "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": 35, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} x & = & \\frac{{x'}}{{x'}^{2} + {y'}^{2}} \\\\ y & = & \\frac{{y'}}{{x'}^{2} + {y'}^{2}} \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} x & = & \\frac{{x'}}{{x'}^{2} + {y'}^{2}} \\\\ y & = & \\frac{{y'}}{{x'}^{2} + {y'}^{2}} \\end{array}\\right.$" ], "text/plain": [ "x = xp/(xp^2 + yp^2)\n", "y = yp/(xp^2 + yp^2)" ] }, "execution_count": 35, "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": 36, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left(U,(x, y)\\right), \\left(V,({x'}, {y'})\\right), \\left(W,(x, y)\\right), \\left(W,({x'}, {y'})\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left(U,(x, y)\\right), \\left(V,({x'}, {y'})\\right), \\left(W,(x, y)\\right), \\left(W,({x'}, {y'})\\right)\\right]$" ], "text/plain": [ "[Chart (U, (x, y)),\n", " Chart (V, (xp, yp)),\n", " Chart (W, (x, y)),\n", " Chart (W, (xp, yp))]" ] }, "execution_count": 36, "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": 37, "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": 38, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(W,(x, y)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(W,(x, y)\\right)$" ], "text/plain": [ "Chart (W, (x, y))" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN_W = stereoN.restrict(W)\n", "stereoN_W" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN_W is S2.atlas()[2]" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(W,({x'}, {y'})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(W,({x'}, {y'})\\right)$" ], "text/plain": [ "Chart (W, (xp, yp))" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS_W = stereoS.restrict(W)\n", "stereoS_W" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Coordinate charts are endoved with a method `plot`. For instance, \n", "we may plot the chart $(W, (x',y'))$ in terms of itself, as a grid:" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 18 graphics primitives" ] }, "execution_count": 41, "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 $(x',y')=(0,0)$ and\n", "ask for the coordinate lines along which $x'$ (resp. $y'$) varies to be colored in purple (resp. cyan):" ] }, { "cell_type": "code", "execution_count": 42, "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": [ "graph = (stereoS_W.plot(stereoN, ranges={xp:[-6,-0.02], yp:[-6,-0.02]},\n", " color={xp: 'purple', yp: 'cyan'}) \n", " + stereoS_W.plot(stereoN, ranges={xp:[-6,-0.02], yp:[0.02,6]},\n", " color={xp: 'purple', yp: 'cyan'})\n", " + stereoS_W.plot(stereoN, ranges={xp:[0.02,6], yp:[-6,-0.02]},\n", " color={xp: 'purple', yp: 'cyan'})\n", " + stereoS_W.plot(stereoN, ranges={xp:[0.02,6], yp:[0.02,6]},\n", " color={xp: 'purple', yp: 'cyan'}))\n", "graph.show(xmin=-1.5, xmax=1.5, ymin=-1.5, ymax=1.5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Spherical coordinates\n", "\n", "The standard **spherical 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": 43, "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": 44, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(A,(x, y)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(A,(x, y)\\right)$" ], "text/plain": [ "Chart (A, (x, y))" ] }, "execution_count": 44, "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": 45, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(A,({\\theta}, {\\phi})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(A,({\\theta}, {\\phi})\\right)$" ], "text/plain": [ "Chart (A, (th, ph))" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher. = A.chart(r'th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n", "spher" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right)\$" ], "text/latex": [ "$\\displaystyle {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right)$" ], "text/plain": [ "th: (0, pi); ph: (0, 2*pi)" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher.coord_range()" ] }, { "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": 47, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\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.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} x & = & -\\frac{\\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right)}{\\cos\\left({\\theta}\\right) - 1} \\\\ y & = & -\\frac{\\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right)}{\\cos\\left({\\theta}\\right) - 1} \\end{array}\\right.$" ], "text/plain": [ "x = -cos(ph)*sin(th)/(cos(th) - 1)\n", "y = -sin(ph)*sin(th)/(cos(th) - 1)" ] }, "execution_count": 47, "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": 48, "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": 49, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\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.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} {\\theta} & = & 2 \\, \\arctan\\left(\\frac{1}{\\sqrt{x^{2} + y^{2}}}\\right) \\\\ {\\phi} & = & \\pi + \\arctan\\left(-y, -x\\right) \\end{array}\\right.$" ], "text/plain": [ "th = 2*arctan(1/sqrt(x^2 + y^2))\n", "ph = pi + arctan2(-y, -x)" ] }, "execution_count": 49, "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 $(A,(x,y))\\rightarrow (A,(x',y'))$ via the operator `*`:" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\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.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} {x'} & = & -\\frac{\\cos\\left({\\phi}\\right) \\cos\\left({\\theta}\\right) - \\cos\\left({\\phi}\\right)}{\\sin\\left({\\theta}\\right)} \\\\ {y'} & = & -\\frac{\\cos\\left({\\theta}\\right) \\sin\\left({\\phi}\\right) - \\sin\\left({\\phi}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}\\right.$" ], "text/plain": [ "xp = -(cos(ph)*cos(th) - cos(ph))/sin(th)\n", "yp = -(cos(th)*sin(ph) - sin(ph))/sin(th)" ] }, "execution_count": 50, "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": 51, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\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.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} {\\theta} & = & 2 \\, \\arctan\\left(\\sqrt{{x'}^{2} + {y'}^{2}}\\right) \\\\ {\\phi} & = & \\pi - \\arctan\\left(\\frac{{y'}}{{x'}^{2} + {y'}^{2}}, -\\frac{{x'}}{{x'}^{2} + {y'}^{2}}\\right) \\end{array}\\right.$" ], "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": 51, "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": 52, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\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]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left(U,(x, y)\\right), \\left(V,({x'}, {y'})\\right), \\left(W,(x, y)\\right), \\left(W,({x'}, {y'})\\right), \\left(A,(x, y)\\right), \\left(A,({x'}, {y'})\\right), \\left(A,({\\theta}, {\\phi})\\right)\\right]$" ], "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": 52, "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": 53, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 30 graphics primitives" ] }, "execution_count": 53, "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", "\n", "To create a point on $\\mathbb{S}^2$, we use SageMath's ***parent / element*** syntax, i.e. the call operator `S2(...)` acting on the parent `S2`, with the point's coordinates in some chart as argument. \n", "\n", "For instance, 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": 54, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point N on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "N = S2((0,0), chart=stereoS, name='N')\n", "print(N)" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point S on the 2-dimensional differentiable manifold S^2\n" ] } ], "source": [ "S = S2((0,0), chart=stereoN, name='S')\n", "print(S)" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathbb{S}^2\$" ], "text/latex": [ "$\\displaystyle \\mathbb{S}^2$" ], "text/plain": [ "2-dimensional differentiable manifold S^2" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N.parent()" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathbb{S}^2\$" ], "text/latex": [ "$\\displaystyle \\mathbb{S}^2$" ], "text/plain": [ "2-dimensional differentiable manifold S^2" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S.parent()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We have of course

" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N in S2" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{False}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{False}$" ], "text/plain": [ "False" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N in U" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N in V" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{False}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{False}$" ], "text/plain": [ "False" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N in A" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us introduce some point $p$ of stereographic coordinates $(x,y) = (1,2)$:" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [], "source": [ "p = S2((1,2), chart=stereoN, name='p')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$p$ lies in the open subset $A$:" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p in A" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Charts acting on points" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "By definition, a chart maps points to pairs of real numbers (the point's coordinates): " ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(1, 2\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(1, 2\\right)$" ], "text/plain": [ "(1, 2)" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoN(p) # by definition of p" ] }, { "cell_type": "code", "execution_count": 65, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\frac{1}{5}, \\frac{2}{5}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\frac{1}{5}, \\frac{2}{5}\\right)$" ], "text/plain": [ "(1/5, 2/5)" ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS(p)" ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(2 \\, \\arctan\\left(\\frac{1}{5} \\, \\sqrt{5}\\right), \\arctan\\left(2\\right)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(2 \\, \\arctan\\left(\\frac{1}{5} \\, \\sqrt{5}\\right), \\arctan\\left(2\\right)\\right)$" ], "text/plain": [ "(2*arctan(1/5*sqrt(5)), arctan(2))" ] }, "execution_count": 66, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher(p)" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(0, 0\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(0, 0\\right)$" ], "text/plain": [ "(0, 0)" ] }, "execution_count": 67, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stereoS(N)" ] }, { "cell_type": "code", "execution_count": 68, "metadata": {}, "outputs": [], "source": [ "#stereoN(N) ## returns an error" ] }, { "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": 69, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\mathbb{R}^3,(X, Y, Z)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\mathbb{R}^3,(X, Y, Z)\\right)$" ], "text/plain": [ "Chart (R^3, (X, Y, Z))" ] }, "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R3. = EuclideanSpace(name='R^3', latex_name=r'\\mathbb{R}^3', metric_name='h')\n", "cartesian = R3.cartesian_coordinates()\n", "cartesian" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As an Euclidean space, `R3` is considered by Sage as a smooth manifold:" ] }, { "cell_type": "code", "execution_count": 70, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Join of Category of smooth manifolds over Real Field with 53 bits of precision and Category of connected manifolds over Real Field with 53 bits of precision and Category of complete metric spaces\n" ] } ], "source": [ "print(R3.category())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The embedding $\\Phi: \\mathbb{S}^2 \\longmapsto \\mathbb{R}^3$ is then defined via the method `diff_map` by providing 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": 71, "metadata": {}, "outputs": [], "source": [ "Phi = S2.diff_map(R3, {(stereoN, cartesian): \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, cartesian): \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": 72, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\text{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) \\\\ \\text{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}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\text{on}\\ U : & \\left(x, y\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, x}{x^{2} + y^{2} + 1}, \\frac{2 \\, y}{x^{2} + y^{2} + 1}, \\frac{x^{2} + y^{2} - 1}{x^{2} + y^{2} + 1}\\right) \\\\ \\text{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, {x'}}{{x'}^{2} + {y'}^{2} + 1}, \\frac{2 \\, {y'}}{{x'}^{2} + {y'}^{2} + 1}, -\\frac{{x'}^{2} + {y'}^{2} - 1}{{x'}^{2} + {y'}^{2} + 1}\\right) \\end{array}$" ], "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": 72, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.display()" ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{Hom}\\left(\\mathbb{S}^2,\\mathbb{R}^3\\right)\$" ], "text/latex": [ "$\\displaystyle \\mathrm{Hom}\\left(\\mathbb{S}^2,\\mathbb{R}^3\\right)$" ], "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": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.parent()" ] }, { "cell_type": "code", "execution_count": 74, "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": 75, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 75, "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": 76, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point Phi(N) on the Euclidean space R^3\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\Phi\\left(N\\right)\$" ], "text/latex": [ "$\\displaystyle \\Phi\\left(N\\right)$" ], "text/plain": [ "Point Phi(N) on the Euclidean space R^3" ] }, "execution_count": 76, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N1 = Phi(N)\n", "print(N1)\n", "N1" ] }, { "cell_type": "code", "execution_count": 77, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(0, 0, 1\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(0, 0, 1\\right)$" ], "text/plain": [ "(0, 0, 1)" ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "cartesian(N1)" ] }, { "cell_type": "code", "execution_count": 78, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point Phi(S) on the Euclidean space R^3\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\Phi\\left(S\\right)\$" ], "text/latex": [ "$\\displaystyle \\Phi\\left(S\\right)$" ], "text/plain": [ "Point Phi(S) on the Euclidean space R^3" ] }, "execution_count": 78, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S1 = Phi(S)\n", "print(S1)\n", "S1" ] }, { "cell_type": "code", "execution_count": 79, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(0, 0, -1\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(0, 0, -1\\right)$" ], "text/plain": [ "(0, 0, -1)" ] }, "execution_count": 79, "metadata": {}, "output_type": "execute_result" } ], "source": [ "cartesian(S1)" ] }, { "cell_type": "code", "execution_count": 80, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point Phi(p) on the Euclidean space R^3\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\Phi\\left(p\\right)\$" ], "text/latex": [ "$\\displaystyle \\Phi\\left(p\\right)$" ], "text/plain": [ "Point Phi(p) on the Euclidean space R^3" ] }, "execution_count": 80, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p1 = Phi(p)\n", "print(p1)\n", "p1" ] }, { "cell_type": "code", "execution_count": 81, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\frac{1}{3}, \\frac{2}{3}, \\frac{2}{3}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\frac{1}{3}, \\frac{2}{3}, \\frac{2}{3}\\right)$" ], "text/plain": [ "(1/3, 2/3, 2/3)" ] }, "execution_count": 81, "metadata": {}, "output_type": "execute_result" } ], "source": [ "cartesian(p1)" ] }, { "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. This triggers a computation involving the transition map $(A,(x,y))\\rightarrow (A,(\\theta,\\phi))$:" ] }, { "cell_type": "code", "execution_count": 82, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\text{on}\\ A : & \\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}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\text{on}\\ A : & \\left(x, y\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, x}{x^{2} + y^{2} + 1}, \\frac{2 \\, y}{x^{2} + y^{2} + 1}, \\frac{x^{2} + y^{2} - 1}{x^{2} + y^{2} + 1}\\right) \\end{array}$" ], "text/plain": [ "Phi: S^2 → R^3\n", "on A: (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))" ] }, "execution_count": 82, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.display(stereoN_A, cartesian)" ] }, { "cell_type": "code", "execution_count": 83, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\text{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}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\text{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), \\cos\\left({\\theta}\\right)\\right) \\end{array}$" ], "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": 83, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.display(spher, cartesian)" ] }, { "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": 84, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 84, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph_spher = spher.plot(chart=cartesian, 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": 85, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 85, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph = stereoN.plot(chart=cartesian, mapping=Phi, number_values=25, \n", " label_axes=False)\n", "graph" ] }, { "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": 86, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "