{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Hyperbolic plane $\\mathbb{H}^2$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*This worksheet illustrates some features of [SageManifolds](http://sagemanifolds.obspm.fr/) (v1.0, as included in SageMath 7.5) on computations regarding the hyperbolic plane.*\n", "\n", "Click [here](https://raw.githubusercontent.com/sagemanifolds/SageManifolds/master/Worksheets/v1.0/SM_hyperbolic_plane.ipynb) to download the worksheet file (ipynb format). To run it, you must start SageMath with the Jupyter notebook, via the command `sage -n jupyter`" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*NB:* a version of SageMath at least equal to 7.5 is required to run this worksheet:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "'SageMath version 7.5, Release Date: 2017-01-11'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we set up the notebook to display mathematical objects using LaTeX formatting:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We also define a viewer for 3D plots (use `'threejs'` or `'jmol'` for interactive 3D graphics):" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "viewer3D = 'threejs' # must be 'threejs', jmol', 'tachyon' or None (default)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We declare $\\mathbb{H}^2$ as a 2-dimensional differentiable manifold:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2-dimensional differentiable manifold H2\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "2-dimensional differentiable manifold H2" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H2 = Manifold(2, 'H2', latex_name=r'\\mathbb{H}^2', start_index=1)\n", "print(H2)\n", "H2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We shall introduce charts on $\\mathbb{H}^2$ that are related to various models of the hyperbolic plane as submanifolds of $\\mathbb{R}^3$. Therefore, we start by declaring $\\mathbb{R}^3$ as a 3-dimensional manifold equiped with a global chart: the chart of Cartesian coordinates $(X,Y,Z)$:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R3, (X, Y, Z))" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R3 = Manifold(3, 'R3', latex_name=r'\\mathbb{R}^3', start_index=1)\n", "X3. = R3.chart()\n", "X3" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Hyperboloid model\n", "\n", "The first chart we introduce is related to the **hyperboloid model of $\\mathbb{H}^2$**, namely to the representation of $\\mathbb{H}^2$ as the upper sheet ($Z>0$) of the hyperboloid of two sheets defined in $\\mathbb{R}^3$ by the equation $X^2 + Y^2 - Z^2 = -1$:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (H2, (X, Y))" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_hyp. = H2.chart()\n", "X_hyp" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The corresponding embedding of $\\mathbb{H}^2$ in $\\mathbb{R}^3$ is" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Phi_1: H2 --> R3\n", " (X, Y) |--> (X, Y, Z) = (X, Y, sqrt(X^2 + Y^2 + 1))" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi1 = H2.diff_map(R3, [X, Y, sqrt(1+X^2+Y^2)], name='Phi_1', latex_name=r'\\Phi_1')\n", "Phi1.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "By plotting the chart $\\left(\\mathbb{H}^2,(X,Y)\\right)$ in terms of the Cartesian coordinates of $\\mathbb{R}^3$, we get a graphical view of $\\Phi_1(\\mathbb{H}^2)$:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(X_hyp.plot(X3, mapping=Phi1, number_values=15, color='blue'), aspect_ratio=1, \n", " viewer=viewer3D, figsize=7)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "A second chart is obtained from the polar coordinates $(r,\\varphi)$ associated with $(X,Y)$. Contrary to $(X,Y)$, the polar chart is not defined on the whole $\\mathbb{H}^2$, but on the complement $U$ of the segment $\\{Y=0, x\\geq 0\\}$: " ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Open subset U of the 2-dimensional differentiable manifold H2\n" ] } ], "source": [ "U = H2.open_subset('U', coord_def={X_hyp: (Y!=0, X<0)})\n", "print(U)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that (y!=0, x<0) stands for $y\\not=0$ OR $x<0$; the condition $y\\not=0$ AND $x<0$ would have been written [y!=0, x<0] instead." ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (U, (r, ph))" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_pol. = U.chart(r'r:(0,+oo) ph:(0,2*pi):\\varphi')\n", "X_pol" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "r: (0, +oo); ph: (0, 2*pi)" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_pol.coord_range()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We specify the transition map between the charts $\\left(U,(r,\\varphi)\\right)$ and $\\left(\\mathbb{H}^2,(X,Y)\\right)$ as $X=r\\cos\\varphi$, $Y=r\\sin\\varphi$:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Change of coordinates from Chart (U, (r, ph)) to Chart (U, (X, Y))" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_hyp = X_pol.transition_map(X_hyp, [r*cos(ph), r*sin(ph)])\n", "pol_to_hyp" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "X = r*cos(ph)\n", "Y = r*sin(ph)" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_hyp.display()" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [], "source": [ "pol_to_hyp.set_inverse(sqrt(X^2+Y^2), atan2(Y, X)) " ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "r = sqrt(X^2 + Y^2)\n", "ph = arctan2(Y, X)" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_hyp.inverse().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The restriction of the embedding $\\Phi_1$ to $U$ has then two coordinate expressions:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Phi_1: U --> R3\n", " (X, Y) |--> (X, Y, Z) = (X, Y, sqrt(X^2 + Y^2 + 1))\n", " (r, ph) |--> (X, Y, Z) = (r*cos(ph), r*sin(ph), sqrt(r^2 + 1))" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi1.restrict(U).display()" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph_hyp = X_pol.plot(X3, mapping=Phi1.restrict(U), number_values=15, ranges={r: (0,3)}, \n", " color='blue')\n", "show(graph_hyp, aspect_ratio=1, viewer=viewer3D, figsize=7)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "{(Chart (H2, (X, Y)),\n", " Chart (R3, (X, Y, Z))): Coordinate functions (X, Y, sqrt(X^2 + Y^2 + 1)) on the Chart (H2, (X, Y))}" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi1._coord_expression" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Metric and curvature\n", "\n", "The metric on $\\mathbb{H}^2$ is that induced by the Minkowksy metric on $\\mathbb{R}^3$: \n", "$$ \\eta = \\mathrm{d}X\\otimes\\mathrm{d}X + \\mathrm{d}Y\\otimes\\mathrm{d}Y\n", " - \\mathrm{d}Z\\otimes\\mathrm{d}Z $$" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "eta = dX*dX + dY*dY - dZ*dZ" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eta = R3.lorentzian_metric('eta', latex_name=r'\\eta')\n", "eta[1,1] = 1 ; eta[2,2] = 1 ; eta[3,3] = -1\n", "eta.display()" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g = (Y^2 + 1)/(X^2 + Y^2 + 1) dX*dX - X*Y/(X^2 + Y^2 + 1) dX*dY - X*Y/(X^2 + Y^2 + 1) dY*dX + (X^2 + 1)/(X^2 + Y^2 + 1) dY*dY" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g = H2.metric('g')\n", "g.set( Phi1.pullback(eta) )\n", "g.display() " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The expression of the metric tensor in terms of the polar coordinates is" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g = 1/(r^2 + 1) dr*dr + r^2 dph*dph" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_pol.frame(), X_pol)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Riemann curvature tensor associated with $g$ is" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field Riem(g) of type (1,3) on the 2-dimensional differentiable manifold H2\n" ] } ], "source": [ "Riem = g.riemann()\n", "print(Riem)" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Riem(g) = -r^2 d/dr*dph*dr*dph + r^2 d/dr*dph*dph*dr + 1/(r^2 + 1) d/dph*dr*dr*dph - 1/(r^2 + 1) d/dph*dr*dph*dr" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem.display(X_pol.frame(), X_pol)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Ricci tensor and the Ricci scalar:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms Ric(g) on the 2-dimensional differentiable manifold H2\n" ] } ], "source": [ "Ric = g.ricci()\n", "print(Ric)" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Ric(g) = -1/(r^2 + 1) dr*dr - r^2 dph*dph" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric.display(X_pol.frame(), X_pol)" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field r(g) on the 2-dimensional differentiable manifold H2\n" ] } ], "source": [ "Rscal = g.ricci_scalar()\n", "print(Rscal)" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "r(g): H2 --> R\n", " (X, Y) |--> -2\n", "on U: (r, ph) |--> -2" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Rscal.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Hence we recover the fact that $(\\mathbb{H}^2,g)$ is a space of **constant negative curvature**." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In dimension 2, the Riemann curvature tensor is entirely determined by the Ricci scalar $R$ according to\n", "\n", "$$R^i_{\\ \\, jlk} = \\frac{R}{2} \\left( \\delta^i_{\\ \\, k} g_{jl} - \\delta^i_{\\ \\, l} g_{jk} \\right)$$\n", "\n", "Let us check this formula here, under the form $R^i_{\\ \\, jlk} = -R g_{j[k} \\delta^i_{\\ \\, l]}$:" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "True" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "delta = H2.tangent_identity_field()\n", "Riem == - Rscal*(g*delta).antisymmetrize(2,3) # 2,3 = last positions of the type-(1,3) tensor g*delta " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Similarly the relation $\\mathrm{Ric} = (R/2)\\; g$ must hold:" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "True" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric == (Rscal/2)*g" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Poincaré disk model\n", "\n", "The Poincaré disk model of $\\mathbb{H}^2$ is obtained by stereographic projection from the point $S=(0,0,-1)$ of the hyperboloid model to the plane $Z=0$. The radial coordinate $R$ of the image of a point of polar coordinate $(r,\\varphi)$ is\n", "$$ R = \\frac{r}{1+\\sqrt{1+r^2}}.$$\n", "Hence we define the Poincaré disk chart on $\\mathbb{H}^2$ by" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (U, (R, ph))" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_Pdisk. = U.chart(r'R:(0,1) ph:(0,2*pi):\\varphi')\n", "X_Pdisk" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "R: (0, 1); ph: (0, 2*pi)" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_Pdisk.coord_range()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and relate it to the hyperboloid polar chart by" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Change of coordinates from Chart (U, (r, ph)) to Chart (U, (R, ph))" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_Pdisk = X_pol.transition_map(X_Pdisk, [r/(1+sqrt(1+r^2)), ph])\n", "pol_to_Pdisk" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "R = r/(sqrt(r^2 + 1) + 1)\n", "ph = ph" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_Pdisk.display()" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "r = -2*R/(R^2 - 1)\n", "ph = ph" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_Pdisk.set_inverse(2*R/(1-R^2), ph)\n", "pol_to_Pdisk.inverse().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A view of the Poincaré disk chart via the embedding $\\Phi_1$:" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(X_Pdisk.plot(X3, mapping=Phi1.restrict(U), ranges={R: (0,0.9)}, color='blue',\n", " number_values=15), aspect_ratio=1, viewer=viewer3D, figsize=7)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The expression of the metric tensor in terms of coordinates $(R,\\varphi)$:" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g = 4/(R^4 - 2*R^2 + 1) dR*dR + 4*R^2/(R^4 - 2*R^2 + 1) dph*dph" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_Pdisk.frame(), X_Pdisk)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We may factorize each metric component:" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g = 4/((R + 1)^2*(R - 1)^2) dR*dR + 4*R^2/((R + 1)^2*(R - 1)^2) dph*dph" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "for i in [1,2]:\n", " g[X_Pdisk.frame(), i, i, X_Pdisk].factor()\n", "g.display(X_Pdisk.frame(), X_Pdisk)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Cartesian coordinates on the Poincaré disk\n", "\n", "Let us introduce Cartesian coordinates $(u,v)$ on the Poincaré disk; since the latter has a unit radius, this amounts to define the following chart on $\\mathbb{H}^2$:" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (H2, (u, v))" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_Pdisk_cart. = H2.chart('u:(-1,1) v:(-1,1)')\n", "X_Pdisk_cart.add_restrictions(u^2+v^2 < 1)\n", "X_Pdisk_cart" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On $U$, the Cartesian coordinates $(u,v)$ are related to the polar coordinates $(R,\\varphi)$ by the standard formulas:" ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Change of coordinates from Chart (U, (R, ph)) to Chart (U, (u, v))" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_to_Pdisk_cart = X_Pdisk.transition_map(X_Pdisk_cart, [R*cos(ph), R*sin(ph)])\n", "Pdisk_to_Pdisk_cart" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "u = R*cos(ph)\n", "v = R*sin(ph)" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_to_Pdisk_cart.display()" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "R = sqrt(u^2 + v^2)\n", "ph = arctan2(v, u)" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_to_Pdisk_cart.set_inverse(sqrt(u^2+v^2), atan2(v, u)) \n", "Pdisk_to_Pdisk_cart.inverse().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The embedding of $\\mathbb{H}^2$ in $\\mathbb{R}^3$ associated with the Poincaré disk model is naturally defined as" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Phi_2: H2 --> R3\n", " (u, v) |--> (X, Y, Z) = (u, v, 0)" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi2 = H2.diff_map(R3, {(X_Pdisk_cart, X3): [u, v, 0]},\n", " name='Phi_2', latex_name=r'\\Phi_2')\n", "Phi2.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us use it to draw the Poincaré disk in $\\mathbb{R}^3$:" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph_disk_uv = X_Pdisk_cart.plot(X3, mapping=Phi2, number_values=15)\n", "show(graph_disk_uv, viewer=viewer3D, figsize=7)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On $U$, the change of coordinates $(r,\\varphi) \\rightarrow (u,v)$ is obtained by combining the changes $(r,\\varphi) \\rightarrow (R,\\varphi)$ and $(R,\\varphi) \\rightarrow (u,v)$:" ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Change of coordinates from Chart (U, (r, ph)) to Chart (U, (u, v))" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_Pdisk_cart = Pdisk_to_Pdisk_cart * pol_to_Pdisk\n", "pol_to_Pdisk_cart" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "u = r*cos(ph)/(sqrt(r^2 + 1) + 1)\n", "v = r*sin(ph)/(sqrt(r^2 + 1) + 1)" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_Pdisk_cart.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Still on $U$, the change of coordinates $(X,Y) \\rightarrow (u,v)$ is obtained by combining the changes $(X,Y) \\rightarrow (r,\\varphi)$ with $(r,\\varphi) \\rightarrow (u,v)$:" ] }, { "cell_type": "code", "execution_count": 46, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Change of coordinates from Chart (U, (X, Y)) to Chart (U, (u, v))" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hyp_to_Pdisk_cart_U = pol_to_Pdisk_cart * pol_to_hyp.inverse()\n", "hyp_to_Pdisk_cart_U" ] }, { "cell_type": "code", "execution_count": 47, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "u = X/(sqrt(X^2 + Y^2 + 1) + 1)\n", "v = Y/(sqrt(X^2 + Y^2 + 1) + 1)" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hyp_to_Pdisk_cart_U.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We use the above expression to extend the change of coordinates $(X,Y) \\rightarrow (u,v)$ from $U$ to the whole manifold $\\mathbb{H}^2$:" ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Change of coordinates from Chart (H2, (X, Y)) to Chart (H2, (u, v))" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hyp_to_Pdisk_cart = X_hyp.transition_map(X_Pdisk_cart, hyp_to_Pdisk_cart_U(X,Y))\n", "hyp_to_Pdisk_cart" ] }, { "cell_type": "code", "execution_count": 49, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "u = X/(sqrt(X^2 + Y^2 + 1) + 1)\n", "v = Y/(sqrt(X^2 + Y^2 + 1) + 1)" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hyp_to_Pdisk_cart.display()" ] }, { "cell_type": "code", "execution_count": 50, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "X = -2*u/(u^2 + v^2 - 1)\n", "Y = -2*v/(u^2 + v^2 - 1)" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hyp_to_Pdisk_cart.set_inverse(2*u/(1-u^2-v^2), 2*v/(1-u^2-v^2))\n", "hyp_to_Pdisk_cart.inverse().display()" ] }, { "cell_type": "code", "execution_count": 51, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph_Pdisk = X_pol.plot(X3, mapping=Phi2.restrict(U), ranges={r: (0, 20)}, number_values=15, \n", " label_axes=False)\n", "show(graph_hyp + graph_Pdisk, aspect_ratio=1, viewer=viewer3D, figsize=7)" ] }, { "cell_type": "code", "execution_count": 52, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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vIXzi4zEpEQnRqpUQt9wixIcfCvH330IUFdXu3DYQXiOAnFFQIMRff0HQTpok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by9BeG992z7DlwIjk+A7t3DhfT3RzkoXcx+/VDaPmsWND83b6LyifKoZHL477/4nfQ7Q4cij+vr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wMJs/HyTnxRfxWc+eEE77+AjuLyV6JTeE8wgQF2kRwd+bnIybsGNHaGG6d4ewa8MGbHP0KML3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\npas3Mkxj1KKF/XP3l1/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9OiF79hCyZQsh33yD9eaddwh5911Cli3DupSZifGh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"text/plain": [ "Graphics object consisting of 30 graphics primitives" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_pol.plot(X_Pdisk_cart, ranges={r: (0, 20)}, number_values=15)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Metric tensor in Poincaré disk coordinates $(u,v)$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "From now on, we are using the Poincaré disk chart $(\\mathbb{H}^2,(u,v))$ as the default one on $\\mathbb{H}^2$:" ] }, { "cell_type": "code", "execution_count": 53, "metadata": { "collapsed": true }, "outputs": [], "source": [ "H2.set_default_chart(X_Pdisk_cart)\n", "H2.set_default_frame(X_Pdisk_cart.frame())" ] }, { "cell_type": "code", "execution_count": 54, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g = (u^4 + v^4 + 2*(u^2 + 1)*v^2 - 2*u^2 + 1)/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) dX*dX - 4*u*v/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) dX*dY - 4*u*v/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) dY*dX + (u^4 + v^4 + 2*(u^2 - 1)*v^2 + 2*u^2 + 1)/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) dY*dY" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_hyp.frame())" ] }, { "cell_type": "code", "execution_count": 55, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g = 4/(u^4 + v^4 + 2*(u^2 - 1)*v^2 - 2*u^2 + 1) du*du + 4/(u^4 + v^4 + 2*(u^2 - 1)*v^2 - 2*u^2 + 1) dv*dv" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display()" ] }, { "cell_type": "code", "execution_count": 56, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g = 4/(u^2 + v^2 - 1)^2 du*du + 4/(u^2 + v^2 - 1)^2 dv*dv" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[1,1].factor() ; g[2,2].factor()\n", "g.display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "## Hemispherical model\n", "\n", "The **hemispherical model of $\\mathbb{H}^2$** is obtained by the inverse stereographic projection from the point $S = (0,0,-1)$ of the Poincaré disk to the unit sphere $X^2+Y^2+Z^2=1$. This induces a spherical coordinate chart on $U$:" ] }, { "cell_type": "code", "execution_count": 57, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (U, (th, ph))" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_spher. = U.chart(r'th:(0,pi/2):\\theta ph:(0,2*pi):\\varphi')\n", "X_spher" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "From the stereographic projection from $S$, we obtain that\n", "\\begin{equation}\n", "\\sin\\theta = \\frac{2R}{1+R^2}\n", "\\end{equation}\n", "Hence the transition map:" ] }, { "cell_type": "code", "execution_count": 58, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Change of coordinates from Chart (U, (R, ph)) to Chart (U, (th, ph))" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_to_spher = X_Pdisk.transition_map(X_spher, [arcsin(2*R/(1+R^2)), ph])\n", "Pdisk_to_spher" ] }, { "cell_type": "code", "execution_count": 59, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "th = arcsin(2*R/(R^2 + 1))\n", "ph = ph" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_to_spher.display()" ] }, { "cell_type": "code", "execution_count": 60, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "R = sin(th)/(cos(th) + 1)\n", "ph = ph" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_to_spher.set_inverse(sin(th)/(1+cos(th)), ph)\n", "Pdisk_to_spher.inverse().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the spherical coordinates $(\\theta,\\varphi)$, the metric takes the following form:" ] }, { "cell_type": "code", "execution_count": 61, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g = cos(th)^(-2) dth*dth + sin(th)^2/cos(th)^2 dph*dph" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_spher.frame(), X_spher)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The embedding of $\\mathbb{H}^2$ in $\\mathbb{R}^3$ associated with the hemispherical model is naturally:" ] }, { "cell_type": "code", "execution_count": 62, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Phi_3: H2 --> R3\n", "on U: (R, ph) |--> (X, Y, Z) = (2*R*cos(ph)/(R^2 + 1), 2*R*sin(ph)/(R^2 + 1), -(R^2 - 1)/(R^2 + 1))\n", "on U: (th, ph) |--> (X, Y, Z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th))" ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi3 = H2.diff_map(R3, {(X_spher, X3): [sin(th)*cos(ph), sin(th)*sin(ph), cos(th)]},\n", " name='Phi_3', latex_name=r'\\Phi_3')\n", "Phi3.display()" ] }, { "cell_type": "code", "execution_count": 63, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\n", "