{ "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 10.5, Release Date: 2024-12-04'" ] }, "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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", "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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", "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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", "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", "