EPAL-v1/ex.I.10.14.ipynb

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    "<table style=\"width:100%\">\n",
    "    <tr>\n",
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    "\n",
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    "# Check geodesics under the cayley transform\n",
    "\n",
    "**Exercise I.10.14**\n",
    "[ [1] ](#[1])\n",
    "$\n",
    "\\newcommand{\\scalar}[3][]{\\left\\langle #2,#3 \n",
    "        \\right\\rangle_{#1}}\n",
    "\\newcommand{\\Space}[3][]{\\mathbb{#2}^{#3}_{#1}}\n",
    "\\newcommand{\\Space}[3][]{\\mathbb{#2}^{#3}_{#1}}\n",
    "  \\newcommand{\\cycle}[3][]{{#1 C^{#2}_{#3}}}\n",
    "  \\newcommand{\\realline}[3][]{#1 R^{#2}_{#3}}\n",
    "  \\newcommand{\\bs}{\\breve{\\sigma}}\n",
    "  \\newcommand{\\zcycle}[3][]{#1 Z^{#2}_{#3}}\n",
    "  \\newcommand{\\SL}[1][2]{\\mathrm{SL}_{#1}(\\Space{R}{})}\n",
    "  \\newcommand{\\rs}{\\sigma_r}\n",
    "  \\newcommand{\\lvec}[1]{\\overrightarrow{#1}}\n",
    "\\newcommand{\\rmi}{\\mathrm{i}}\n",
    "\\newcommand{\\alli}{\\iota}\n",
    "\\newcommand{\\rme}{\\mathrm{e}}\n",
    "\\newcommand{\\rmd}{\\mathrm{d}}\n",
    "\\newcommand{\\rmh}{\\mathrm{j}}\n",
    "\\newcommand{\\rmp}{\\varepsilon}\n",
    "\\newcommand{\\modulus}[2][]{\\left| #2 \\right|_{#1}}\n",
    "$\n",
    "Check that:\n",
    "1. (e,h) Elliptic and hyperbolic Cayley transforms send the\n",
    "    respective geodesics\n",
    "    passing $\\alli$ to the straight line passing the origin.\n",
    "    Consequently, any geodesics is a cycle orthogonal to the boundary of the unit disk. The\n",
    "    respective invariant metrics on the unit disks are, cf. [Yaglom79, Table~VI],\n",
    "\\begin{equation}\n",
    "      \\sin_{\\rs}^{-1}  \\frac{|w-w'|_\\sigma}{2 \\sqrt{(1+\\sigma w\\bar{w})(1+\\sigma w'\\bar{w}')}},\n",
    "\\end{equation}\n",
    "    where $\\rs=1$ in the elliptic case and has a value depending on\n",
    "    the degree of\n",
    "    space- or light-likeness in the hyperbolic case.\n",
    "2. (p) The $\\bs$-parabolic Cayley\n",
    "    transform maps the\n",
    "    $\\rs$-geodesics passing $\\rmp$ to\n",
    "    the parabolas passing the origin:\n",
    "\\begin{equation}\n",
    "      (\\rs-4\\bs+4t^2)u^2-8tu-4v=0.\n",
    "\\end{equation}\n",
    "    The invariant $\\rs$-metric on the $\\bs$-parabolic unit circle is\n",
    "\\begin{equation}\n",
    "      \\sin_{\\rs }^{-1}  \\frac{|u-u'|}{\\sqrt{(1+v+\\bs u^2)(1+v'+\\bs u'^2)}\n",
    "      }. \n",
    "\\end{equation}"
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    "**Solution.** Make the check for elliptic and hyperbolic geodesics from $\\rmi$"
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    "from init_cycle import *\n",
    "Ceh=cycle2D(1,[t,0],sign,e)\n",
    "print(\"Elliptic and hyperbolic geodesic on unit disk from the centre are straight lines: %s\" %\\\n",
    "bool(cayley(Ceh,sign,0).subs(pow(sign,2)==1,subs_options.algebraic).is_linear()))"
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    "The parabolic geodesics are obtained by the rotation of the main one"
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    "Cp=cycle2D(sign2,[0,2],4,e)\n",
    "Latex(f\"${Cp.sl2_similarity([1,0,t,1],e).string()}$\")"
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    "Latex(\"Geodesics under the parabolic Cayley transform: $%s$\" % \\\n",
    "cayley(Cp.sl2_similarity([1,0,t,1],e),0,sign1).string())"
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    "Latex(\"Geodesics under the parabolic Cayley transform second form: $%s$\" % \\\n",
    "cayley(Cp.sl2_similarity([1,0,t,1],e),0,sign1).string())"
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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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