EPAL-v1/ex.I.10.9.ipynb

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    "<table style=\"width:100%\">\n",
    "    <tr>\n",
    "        <td>\n",
    "            <a href=\"http://v-v-kisil.scienceontheweb.net/MoebInv-notebooks/EPAL-v1/ex.I.10.9.html\">\n",
    "  <img src=\"../svg/view-html.svg\" alt=\"Open In Colab\"/>\n",
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    "        <td>\n",
    "<a href=\"https://github.com/vvkisil/MoebInv-notebooks/blob/master/EPAL-v1/ex.I.10.9.ipynb\">\n",
    "  <img src=\"../svg/open-github.svg\" alt=\"Open In Colab\"/>\n",
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    "        <td>\n",
    "            <a href=\"https://colab.research.google.com/github/vvkisil/MoebInv-notebooks/blob/master/EPAL-v1/ex.I.10.9.ipynb\">\n",
    "  <img src=\"../svg/exec-colab.svg\" alt=\"Open In Colab\"/>\n",
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    "        </td>\n",
    "        <td>\n",
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    "\n",
    "<a  align=\"center\" href=\"https://colab.research.google.com/github/vvkisil/MoebInv-notebooks/blob/master/Introduction/Software_installation_GUI_integration.ipynb\">Google CoLab software installation</a>\n"
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    "# We verify the expression of the parabolic Cayley transform on the cycles\n",
    "\n",
    "**Exercise I.10.9** [ [1] ](#[1])\n",
    "  Let $\\newcommand{\\Space}[3][]{\\mathbb{#2}^{#3}_{#1}}\n",
    "  \\newcommand{\\cycle}[3][]{{#1 C^{#2}_{#3}}}\n",
    "  \\newcommand{\\realline}[3][]{#1 R^{#2}_{#3}}\n",
    "  \\newcommand{\\bs}{\\breve{\\sigma}}$\n",
    "  $\\cycle{s}{a}$ be a cycle in $\\Space[\\sigma]{R}{}$. Check that:\n",
    "\n",
    "  + (e,h) In the elliptic or hyperbolic cases, the Cayley\n",
    "    transform of the cycle $\\cycle{}{\\sigma}$ is\n",
    "    $\\realline{}{\\sigma}\\cycle[\\hat]{}{\\sigma\n",
    "    }\\cycle{}{\\sigma}\\cycle[\\hat]{}{\\sigma }\\realline{}{\\sigma}$,\n",
    "    i.e. the composition of the\n",
    "    similarity (I.6.10) by the cycle\n",
    "    $\\cycle[\\hat]{s}{\\sigma }=(\\sigma ,0,1,1)$ and the similarity by\n",
    "    the real line (see the first and last drawings in\n",
    "    Fig.I.10.2.\n",
    "\n",
    "  + (p) In the parabolic case, the Cayley transform maps a cycle\n",
    "    $(k,l,n,m)$ to the cycle $(k-2\\bs n, l, n ,m-2 n)$.\n",
    "\n",
    "**Hint**\n",
    "    We can follow a similar path to the proof of\n",
    "    Theorem I.4.13. Alternatively, for the first part,\n",
    "    we notice that the matrix $Y_\\sigma $ of the Cayley\n",
    "    transform and the FSCc matrix of the cycle $\\cycle[\\hat]{s}{\\sigma }$\n",
    "    are different by a constant factor. The reflection in the real line\n",
    "    compensates the effect of complex conjugation in the\n",
    "    similarity (I.6.10). "
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    "**Solution.**  Take the Cayley transform of a point..."
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   "source": [
    "from init_cycle import *\n",
    "P0=(u+I*v-I)/(-I*(u+I*v)+1)"
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    "... and its Möbius transformation"
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    "Pc=clifford_moebius_map(one, -e1, sign1*e1, one, P, e)"
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    "Compare both outputs:"
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    "print(\"Clifford calculations coinsides with complex numbers: %s\" %\\\n",
    "((P0+P0.conjugate()-2*Pc[0]).subs({sign : -1,sign1 : -1}).normal().is_zero() and\n",
    "(P0-P0.conjugate()-2*I*Pc[1]).subs({sign : -1,sign1 : -1}).normal().is_zero()))"
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    "\n",
    "This notebook is a part of the *MoebInv notebooks* project [ [2] ](#[2]).\n",
    "\n",
    "### References\n",
    "<a id=\"references\"></a>\n",
    "<a id=\"[1]\"></a>\n",
    "1. Vladimir V. Kisil. *Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of $SL_2(\\mathbb{R})$*. [Imperial College Press](https://www.worldscientific.com/worldscibooks/10.1142/p835), London, 2012. Includes a live DVD.\n",
    "\n",
    "   <a id=\"[2]\"></a>\n",
    "2.  Vladimir V. Kisil, [*MoebInv notebooks*](https://github.com/vvkisil/MoebInv-notebooks), 2019.\n",
    "\n",
    "[Back to Folder](https://github.com/vvkisil/MoebInv-notebooks/blob/master/EPAL-v1/)\n"
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