{ "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 10.3, Release Date: 2024-03-19'" ] }, "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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", "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", "