EPAL-v1/ex.I.5.21.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.5.21.html\">\n",
    "  <img src=\"../svg/view-html.svg\" alt=\"Open In Colab\"/>\n",
    "            </a>\n",
    "        </td>\n",
    "        <td>\n",
    "<a href=\"https://github.com/vvkisil/MoebInv-notebooks/blob/master/EPAL-v1/ex.I.5.21.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.5.21.ipynb\">\n",
    "  <img src=\"../svg/exec-colab.svg\" alt=\"Open In Colab\"/>\n",
    "            </a>\n",
    "        </td>\n",
    "        <td>\n",
    "            <a href=\"https://codeocean.com/capsule/7952650/tree\">\n",
    "  <img src=\"../svg/code-ocean.svg\" alt=\"Code Ocean\"/>\n",
    "            </a>\n",
    "        </td>\n",
    "    </tr>\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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    "# Power of point through the cycle inner product \n",
    "\n",
    "**Exercise I.5.21** [ [1] ](#[1])\n",
    "$\n",
    "\\newcommand{\\scalar}[3][]{\\left\\langle #2,#3 \n",
    "        \\right\\rangle_{#1}}\n",
    "\\newcommand{\\Space}[3][]{\\mathbb{#2}^{#3}_{#1}}\n",
    "\\newcommand{\\Space}[3][]{\\mathbb{#2}^{#3}_{#1}}\n",
    "  \\newcommand{\\cycle}[3][]{{#1 C^{#2}_{#3}}}\n",
    "  \\newcommand{\\realline}[3][]{#1 R^{#2}_{#3}}\n",
    "  \\newcommand{\\bs}{\\breve{\\sigma}}\n",
    "  \\newcommand{\\zcycle}[3][]{#1 Z^{#2}_{#3}}\n",
    "  \\newcommand{\\SL}[1][2]{\\mathrm{SL}_{#1}(\\Space{R}{})}\n",
    "  \\newcommand{\\rs}{\\sigma_r}\n",
    "  \\newcommand{\\lvec}[1]{\\overrightarrow{#1}}\n",
    "\\newcommand{\\rmi}{\\mathrm{i}}\n",
    "\\newcommand{\\alli}{\\iota}\n",
    "\\newcommand{\\rme}{\\mathrm{e}}\n",
    "\\newcommand{\\rmd}{\\mathrm{d}}\n",
    "\\newcommand{\\rmh}{\\mathrm{j}}\n",
    "\\newcommand{\\rmp}{\\varepsilon}\n",
    "\\newcommand{\\modulus}[2][]{\\left| #2 \\right|_{#1}}\n",
    "\\newcommand{\\sperp}{⋋}\n",
    "$\n",
    "  Let $\\zcycle{}{\\sigma}$ be the zero-radius\n",
    "  cycle defined by $z=u-\\alli\\sigma v$,\n",
    "  that is, with the $\\sigma$-centre at the point $(u, v)$.  Show\n",
    "  that, in the elliptic and hyperbolic (but not parabolic) cases, a\n",
    "  power of a point, $(u,v)\\in\\Space[\\sigma]{R}{}$ with\n",
    "  respect to a cycle $\\cycle{}{\\sigma }$, is equal to the cycle\n",
    "  product $\\scalar{\\zcycle{}{\\sigma }}{\\cycle{}{\\sigma }}$, where\n",
    "  both cycles are $k$-normalised."
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    "**Solution.** We will verify it in a loop:"
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    "from init_cycle import *\n",
    "case=[\"E\", \"P\", \"H\"]\n",
    "Zn=cycle2D([u,-sign1*v],e,0)\n",
    "for i in range(0,3,2):\n",
    "    print(\"Power of point is inner product in %s case:  %s\" % \\\n",
    "    (case[i], (C.val(P)-C.cycle_product(Zn,e).subs({sign1 : sign})).subs({sign : i-1}).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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