Hopf-torus-Viviani-curve.ipynb

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   "source": [
    "## <center> The Hopf torus as the preimage  of the Viviani's curve </center>"
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    "The common Viviani's curve on the unit sphere is defined as the intersection between the sphere $x^2+y^2+z^2=1$  and the cylinder $(x-1/2)^2+y^2=1/4$. If the sphere is parameterized by longitude and latitude:\n",
    "$$\\begin{array}{lll} x&=&\\cos(\\theta)\\cos(\\varphi)\\\\\n",
    "y&=&\\sin(\\theta)\\cos(\\varphi), \\quad \\theta\\in (-\\pi, \\pi], \\varphi\\in[-\\pi/2, \\pi/2]\\\\\n",
    "z&=&\\sin(\\varphi),\\end{array}$$\n",
    "then the intersection curve has the parameterization derived\n",
    "from the sphere parameterization, with $\\theta=\\varphi$.\n",
    "Replacing x,y,z in the cylinder equation we get that $\\cos(\\theta)=\\cos(\\varphi)$, i.e $\\theta=\\varphi$. Hence the Viviani's curve has the parameterization:\n",
    "$$\\begin{array}{lll}x&=&\\cos^2(\\theta)\\\\\n",
    "    y&=&0.5\\sin(2\\theta), \\quad \\theta\\in (-\\pi, \\pi]\\\\\n",
    "    z&=&\\sin(\\theta)\\end{array}$$\n",
    "Taking into account that $cos(\\theta)=1-2\\sin^2(\\theta/2)$\n",
    "we get the following parameterization of the Viviani's curve, where $t/2=\\theta$, \n",
    "        $t\\in(-2\\pi, 2\\pi]$:\n",
    "\n",
    "$$\\begin{array}{lll}\n",
    "    x&=&0.5(1+\\cos(t))\\\\\n",
    "    y&=&0.5\\sin(t)\\\\\n",
    "    z&=&\\sin(t/2)\n",
    "    \\end{array}$$"
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    "To get the Hopf torus as the preimage  through the Hopf map, $h:\\mathbb{S}^3\\to\\mathbb{S}^2$,  $h(q)=q*k*q^{-1}$, we rotate this version of the Viviani's curve, about y-axis, with  an angle of $\\pi/2$.\n",
    "After   such a rotation the new Viviani's curve is the intersection\n",
    " of the unit sphere to the cylinder $(x^2+(z+1/2)^2=1/4$, and it is parameterized by:\n",
    "\n",
    "$$\\begin{array}{lll}x&=&\\sin(t/2)\\\\\n",
    "      y&=&0.5\\sin(t)\\quad t\\in(-2\\pi, 2\\pi]\\\\\n",
    "    z&=&-0.5(1+\\cos(t))\\end{array}$$"
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   "source": [
    "![viviani def](https://github.com/empet/Datasets/blob/master/Images/viviani-intersection.png?raw=true)"
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   "source": [
    "The preimage of the Viviani's curve through the  Hopf map is a Hopf torus consisting in two tori,\n",
    "each one  coresponding to a loop in the auto-intersecting Viviani's curve:"
   ]
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   "source": [
    "%%html\n",
    "<img src=\"https://github.com/empet/Datasets/blob/master/Images/viviani-hopftor.gif?raw=true\">"
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   "source": [
    "The fibres corresponding to 40 points on this version of Viviani's curve:"
   ]
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   "source": [
    "![hopf fibs viviani](https://github.com/empet/Datasets/blob/master/Images/Hopf-fibs-viviani.png?raw=true)"
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