EPAL-v1/ex.I.6.5.i.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.6.5.i.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.6.5.i.ipynb\">\n",
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    "            <a href=\"https://colab.research.google.com/github/vvkisil/MoebInv-notebooks/blob/master/EPAL-v1/ex.I.6.5.i.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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    "# Check the geometric orthogonality condition for circles and hyperbolas\n",
    "\n",
    "**Exercise I.6.5.i** [ [1] ](#[1]) Prove \n",
    "    the converses of the two statements in Exercise I.6.3:\n",
    "$\n",
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    "$\n",
    "1. (e,h) Let $\\sigma=\\pm 1$. Then, two cycles in\n",
    "    $\\Space[\\sigma]{R}{}$ (that is, circles or hyperbolas) are\n",
    "    $\\sigma$-orthogonal if slopes $S_1$ and $S_2$ of their\n",
    "    tangents at the intersection point satisfy the condition\n",
    "    \\begin{equation}\n",
    "      \\label{eq:ortho-slopes-rel}\n",
    "      S_1 S_2=\\sigma. \n",
    "    \\end{equation}\n",
    "    The geometrical meaning of this condition can be given either in terms\n",
    "    of angles (A) or centres (C):\n",
    "\n",
    "  + (A) For the case $\\sigma=-1$ (circles),\n",
    "      equation implies orthogonality of the\n",
    "      tangents. For\n",
    "      $\\sigma=1$, two hyperbolas are h-orthogonal if lines with the\n",
    "      slopes $\\pm 1$ bisect the angle of intersection of the\n",
    "      hyperbolas.\n",
    "      \n",
    "\n",
    "  + (C) In the cases $\\sigma=\\pm 1$, the tangent to one cycle\n",
    "      at the intersection point passes the centre of another cycle.\n",
    "      \n",
    "2. (p) Let $\\sigma=0$ and a parabola $\\cycle{}{p}$ have two\n",
    "    real roots $u_1$ and $u_2$. If a parabola\n",
    "    $\\cycle[\\tilde]{}{p}$ is $p$-orthogonal to $\\cycle{}{p}$,\n",
    "    then the tangent to $\\cycle[\\tilde]{}{p}$ at a point above one\n",
    "    of the roots $u_{1,2}$ passes the p-centre\n",
    "    $(\\frac{u_1+u_2}{2},0)$ of $\\cycle{}{p}$.       "
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    "**Solution.** Define a cycle passing point $(u,v)$ "
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    "from init_cycle import *\n",
    "C3=C.subject_to(C.passing(P))"
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    "Define a cycle passing the same point and being orthogonal to `C3`"
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   "source": [
    "C4=C1.subject_to([C1.passing(P), \\\n",
    "(C3.val(P1).diff(u1)*C1.val(P1).diff(u1) \\\n",
    "-sign*C3.val(P1).diff(v1)*C1.val(P1).diff(v1)).subs({v1 : v, u1 : u})==0])"
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    "\"Slopes at the intersection point ensure orthogonality of circles and hyperbolas: %s\" %\\\n",
    "C3.cycle_product(C4).subs({pow(sign,2) : 1}, subs_options.algebraic).is_zero()"
   ]
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   "source": [
    "Check the geometric orthogonality condition for parabolas. A cycle passing the point $(u,v)$"
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   "source": [
    "C5=C1.subject_to(C1.passing(P))"
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   "source": [
    "Its slope at the point $(u,v)$"
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   "source": [
    "Sl5=(C5.val(P1).diff(u1)/C5.val(P1).diff(v1)).subs({u1 : u, v1 : v, sign : 0})"
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   "source": [
    "Diameter of the parabols"
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   "source": [
    "X=v/Sl5"
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   "source": [
    "Parabola having root at $(u,0)$ and the tangent passing its centre"
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    "C6=cycle2D([u+X,0],e.subs({sign : 0}),X*X)\n",
    "\"Parabolas are orthogonal if the tangent above root pass the centre: %s\" % \\\n",
    "bool(C6.is_orthogonal(C5).subs({sign : 0}))"
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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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