EPAL-v1/ex.I.7.7.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.7.7.html\">\n",
    "  <img src=\"../svg/view-html.svg\" alt=\"Open In Colab\"/>\n",
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    "<a href=\"https://github.com/vvkisil/MoebInv-notebooks/blob/master/EPAL-v1/ex.I.7.7.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.7.7.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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    "# First we define the distance of functions\n",
    "\n",
    "**Exercise I.7.7**[ [1] ](#[1])\n",
    "$\n",
    "\\newcommand{\\scalar}[3][]{\\left\\langle #2,#3 \n",
    "        \\right\\rangle_{#1}}\n",
    "\\newcommand{\\Space}[3][]{\\mathbb{#2}^{#3}_{#1}}\n",
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    "  \\newcommand{\\cycle}[3][]{{#1 C^{#2}_{#3}}}\n",
    "  \\newcommand{\\realline}[3][]{#1 R^{#2}_{#3}}\n",
    "  \\newcommand{\\bs}{\\breve{\\sigma}}\n",
    "  \\newcommand{\\zcycle}[3][]{#1 Z^{#2}_{#3}}\n",
    "  \\newcommand{\\SL}[1][2]{\\mathrm{SL}_{#1}(\\Space{R}{})}\n",
    "  \\newcommand{\\rs}{\\sigma_r}\n",
    "  \\newcommand{\\lvec}[1]{\\overrightarrow{#1}}\n",
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    "\\newcommand{\\modulus}[2][]{\\left| #2 \\right|_{#1}}\n",
    "\\newcommand{\\sperp}{⋋}\n",
    "$\n",
    "  Show that the power of a point $W$ \n",
    "  with respect to a cycle is the\n",
    "  product $d_\\sigma (W,P)\\cdot d_\\sigma (W,P')$ of\n",
    "  $(\\sigma,\\sigma)$-distances, where\n",
    "  $P$ and $P'$ are any two points of the cycle which are collinear with $W$.\n",
    "\n",
    "**Hint.**\n",
    "    Take $P=(u,v)$ and $P'=(u',v')$. Then, any collinear $W$ is\n",
    "    $(tu+(1-t)u',tv+(1-t)v')$ for some\n",
    "    $t\\in\\Space{R}{}$. Furthermore, a simple calculation shows that\n",
    "    $d_\\sigma (y,z)\\cdot d_\\sigma (z,y')=\n",
    "    t(t-1)d_\\sigma^2(y,y')$. The last expression is equal to the power\n",
    "    of $W$ with respect to the cycle---this step can be done by CAS."
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    "**Solution.** Define the function for the distance"
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    "from init_cycle import *\n",
    "def dist2(X,Y,s):\n",
    "    return (X[0]-Y[0])**2-s*(X[1]-Y[1])**2"
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   "source": [
    "Define point P"
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   "source": [
    "P=[u,v]"
   ]
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   "source": [
    "Define point P1"
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   "source": [
    "P1=[u1,v1]"
   ]
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   "source": [
    "Define a new point Pt on the line through (u,v) and (u1,v1)"
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   "source": [
    "Pt=[t*u+(1-t)*u1,t*v+(1-t)*v1]"
   ]
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   "source": [
    "Then create a template of a cycle"
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   "source": [
    "C0=cycle2D(1,[l,n],m,e)"
   ]
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    "Now ask the cycle pass points P and P1"
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   "source": [
    "Cr=C0.subject_to([C0.passing(P), C0.passing(P1)],[l,n,m])"
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   "source": [
    "Calculate the power of Pt with respect to the cycle"
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   "source": [
    "E=Cr.val(Pt).normal()"
   ]
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   "source": [
    "and compare it with the product:"
   ]
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   "source": [
    "\"Power is the product of distances (1st form): %r\" % \\\n",
    "(dist2(P,P1,sign)*t*(t-1)-E).expand().normal().is_zero()"
   ]
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   "source": [
    "\"Power is the product of distances (2nd form): %r\" % \\\n",
    "(dist2(P,Pt,sign)*dist2(Pt,P1,sign)-E**2).expand().normal().is_zero()"
   ]
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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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