EPAL-v1/ex.I.8.2.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.8.2.html\">\n",
    "  <img src=\"../svg/view-html.svg\" alt=\"Open In Colab\"/>\n",
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    "        </td>\n",
    "        <td>\n",
    "<a href=\"https://github.com/vvkisil/MoebInv-notebooks/blob/master/EPAL-v1/ex.I.8.2.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.8.2.ipynb\">\n",
    "  <img src=\"../svg/exec-colab.svg\" alt=\"Open In Colab\"/>\n",
    "            </a>\n",
    "        </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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    "# Check properties of the zero-radius cycle at infinity\n",
    "\n",
    "**Exercise I.8.2** [ [1] ](*[1])\n",
    "   Check the following:\n",
    "1. $Z_{\\infty}$ is the image of the zero-radius cycle\n",
    "    $Z_{(0,0)}=(1,0,0,0)$ at the origin under reflection into the unit cycle $(1,\n",
    "    0,0,-1)$---see blue cycles in Fig.~I.6.4(b)--(d) from  [ [1] ](*[1]). \n",
    "2. The following statements are equivalent:\n",
    "     +  A point $(u,v)\\in\\mathbb{R}_\\sigma$ belongs to the\n",
    "        \\emph{zero-radius cycle} $Z_{(0,0)}$ centred at the\n",
    "       origin.\n",
    "     +  The zero-radius cycle $Z_{(u,v)}$ is\n",
    "       $\\sigma$-orthogonal to the zero-radius cycle $Z_{(0,0)}$.\n",
    "     + The inversion $z\\mapsto \\frac{1}{z}$ in the unit cycle is\n",
    "       singular in the point $(u,v)$.\n",
    "     + The image of $Z_{(u,v)}$ under inversion in the\n",
    "       unit cycle is orthogonal to $Z_{\\infty}$.\n",
    "       \n",
    "   If any one of the above statements is true, we also say that the image\n",
    "   of $(u,v)$ under inversion in the unit cycle belongs to the\n",
    "   zero-radius cycle at infinity.\n",
    "  \n",
    "**Hint.**\n",
    "    These can be easily obtained by direct calculations, even without\n",
    "    a CAS.\n"
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    "**Solution.** We define the required cycles:"
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    "from init_cycle import *\n",
    "Z0=cycle2D([0,0],e,0)\n",
    "Cu=cycle2D(1,[0,0],-1,e)"
   ]
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   "source": [
    "Then we simply output the reuired statements:"
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    "Latex(\"Zero cycle at infinity is the reflection of zero cycle at origin: %s\" % \\\n",
    "Zinf.is_equal(Z0.cycle_similarity(Cu)))"
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    "Latex(f\"$(u,v)$ belongs to zero cycle at the origin if: ${Z0.val([u,v])==0}$\")"
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    "Latex(f\"Zero cycle at $(u,v)$ is orthogonal to zero cycle at the origin if: ${Z.is_orthogonal(Z0,es)}$\")"
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    "Latex(f\"Zero cycle at infinity is orthogonal to reflection of $Z(u,v)$ if: ${Zinf.is_orthogonal(Z.cycle_similarity(Cu))}$\")"
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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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