EPAL-v1/ex.I.7.8.ipynb

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    "        <td>\n",
    "            <a href=\"http://v-v-kisil.scienceontheweb.net/MoebInv-notebooks/EPAL-v1/ex.I.7.8.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.7.8.ipynb\">\n",
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    "        <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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    "# Check the formula for an inversive distance for a generic cycle\n",
    "\n",
    "**Exercise I.7.8** [ [1] ](#[1])\n",
    "$\n",
    "\\newcommand{\\scalar}[3][]{\\left\\langle #2,#3 \n",
    "        \\right\\rangle_{#1}}\n",
    "\\newcommand{\\Space}[3][]{\\mathbb{#2}^{#3}_{#1}}\n",
    "\\newcommand{\\Space}[3][]{\\mathbb{#2}^{#3}_{#1}}\n",
    "  \\newcommand{\\cycle}[3][]{{#1 C^{#2}_{#3}}}\n",
    "  \\newcommand{\\realline}[3][]{#1 R^{#2}_{#3}}\n",
    "  \\newcommand{\\bs}{\\breve{\\sigma}}\n",
    "  \\newcommand{\\zcycle}[3][]{#1 Z^{#2}_{#3}}\n",
    "  \\newcommand{\\SL}[1][2]{\\mathrm{SL}_{#1}(\\Space{R}{})}\n",
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    "\\newcommand{\\modulus}[2][]{\\left| #2 \\right|_{#1}}\n",
    "\\newcommand{\\sperp}{⋋}\n",
    "$\n",
    "  Let two cycles have e-centres $P$ and $P'$ with $\\bs$-radii\n",
    "  $r_{\\bs}$ and $r_{\\bs}'$. Then, the $(\\bs,\\bs)$-power%\n",
    "  \\index{power!cycle, of}%\n",
    "  \\index{point!cycle, of} of one cycle with\n",
    "  respect to another is,\n",
    "  \\begin{equation}\n",
    "    \\frac{d_{\\bs,\\bs}^2(P,P')-r_{\\bs}^2-r_{\\bs}^{\\prime 2}}{2r_{\\bs}\\, r_{\\bs}'}. \n",
    "  \\end{equation}"
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    "**Solution** Define cycles `C2` and `C3` with radii `r` and `r1` respectively"
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    "from init_cycle import *\n",
    "r=possymbol(\"r\")\n",
    "r1=possymbol(\"r1\", \"r_1\")\n",
    "C2=cycle2D([u,v],es,pow(r,2)).normalize_det()\n",
    "C3=cycle2D([u1,v1],es,pow(r1,2)).normalize_det()"
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    "Latex(f\"Cycle C2 is: ${C2.string()}$\")"
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    "Latex(\"Power of a cycle with respect to another: $%s$\" % \\\n",
    "C2.cycle_product(C3,e).normal())"
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   "source": [
    "\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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