EPAL-v1/ex.I.10.12.ipynb

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    "<table style=\"width:100%\">\n",
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    "            <a href=\"http://v-v-kisil.scienceontheweb.net/MoebInv-notebooks/EPAL-v1/ex.I.10.12.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.10.12.ipynb\">\n",
    "  <img src=\"../svg/open-github.svg\" alt=\"Open In Colab\"/>\n",
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    "            <a href=\"https://colab.research.google.com/github/vvkisil/MoebInv-notebooks/blob/master/EPAL-v1/ex.I.10.12.ipynb\">\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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    "# The family of infinitesimal cycles is invariant under the Cayley transform \n",
    "\n",
    "**Exercise I.10.12.** [ [1] ](#[1])\n",
    " Show that images of infinitesimal cycles under the parabolic Cayley\n",
    "  transform are, themselves, infinitesimal\n",
    "  cycles.\n",
    "  \n",
    "**Solution.** First we define some suitable substitutions to simplify algebraic equations later."
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    "from init_cycle import *\n",
    "sign_cube = {pow(sign, 3)  : sign}\n",
    "sign1_cube = {pow(sign1, 3)  : sign1}\n",
    "vp=possymbol(\"vp\")\n",
    "sl2_relation = {c*b  : a*d-1}\n",
    "sl2_relation1 = {a  : (1+b*c)/d}"
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    "The check will be performed by the next procedure. It verifies the expression of the Cayley transform on infinitesimal cycles and evaluates its radius."
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    "def infinitesimal_trans(nval):\n",
    "    C10 = cycle2D(1, [u, n],  pow(u,2)-pow(n,2)*sign1+pow(epsilon,2), e).subs(nval)\n",
    "    C11=cayley(C10,0,sign1)\n",
    "    display(Latex(\"Checking infinitesimal cycle: $%s$\" % C10.string()))\n",
    "    display(Latex(\"  Image under Cayley transform of infinitesimal cycle has radius squared: $%s$\" %\\\n",
    "    C11.radius_sq(es).subs(sl2_relation1, subs_options.algebraic).\\\n",
    "    subs(sign_cube, subs_options.algebraic).series(relational(epsilon, 0, eq), 3).normal()))"
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    "Now we do first check, the first for elliptic focus."
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    "infinitesimal_trans({n : (vp-(pow(pow(vp,2)-pow(epsilon,2)*(sign2-sign1),half)))/(sign2-sign1)})"
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    "The second check is for the parabolic focus, i.e. the vertex of the parabola."
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    "infinitesimal_trans({n : pow(epsilon,2)/2/vp, sign2 : sign1})\n",
    "                                                                  "
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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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