{ "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 9.7, Release Date: 2022-09-19'" ] }, "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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"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 \\\\ \\mbox{on}\\ U : & \\left(x, y\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, x}{x^{2} + y^{2} + 1}, \\frac{2 \\, y}{x^{2} + y^{2} + 1}, \\frac{x^{2} + y^{2} - 1}{x^{2} + y^{2} + 1}\\right) \\\\ \\mbox{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, {x'}}{{x'}^{2} + {y'}^{2} + 1}, \\frac{2 \\, {y'}}{{x'}^{2} + {y'}^{2} + 1}, -\\frac{{x'}^{2} + {y'}^{2} - 1}{{x'}^{2} + {y'}^{2} + 1}\\right) \\end{array}\\)" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\mbox{on}\\ U : & \\left(x, y\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, x}{x^{2} + y^{2} + 1}, \\frac{2 \\, y}{x^{2} + y^{2} + 1}, \\frac{x^{2} + y^{2} - 1}{x^{2} + y^{2} + 1}\\right) \\\\ \\mbox{on}\\ V : & \\left({x'}, {y'}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, {x'}}{{x'}^{2} + {y'}^{2} + 1}, \\frac{2 \\, {y'}}{{x'}^{2} + {y'}^{2} + 1}, -\\frac{{x'}^{2} + {y'}^{2} - 1}{{x'}^{2} + {y'}^{2} + 1}\\right) \\end{array}$" ], "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 \\\\ \\mbox{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": [ "$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\mbox{on}\\ A : & \\left(x, y\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, x}{x^{2} + y^{2} + 1}, \\frac{2 \\, y}{x^{2} + y^{2} + 1}, \\frac{x^{2} + y^{2} - 1}{x^{2} + y^{2} + 1}\\right) \\end{array}$" ], "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 \\\\ \\mbox{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), \\cos\\left({\\theta}\\right)\\right) \\end{array}\\)" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathbb{S}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\mbox{on}\\ A : & \\left({\\theta}, {\\phi}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), \\cos\\left({\\theta}\\right)\\right) \\end{array}$" ], "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", "