Conformal-mapping-of-a-tilling- onto-torus.ipynb

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   "source": [
    "## <center> Conformal mapping of a planar tilling  onto torus </center>"
   ]
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  {
   "cell_type": "code",
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   "source": [
    "using PlotlyJS, Images"
   ]
  },
  {
   "cell_type": "code",
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   "source": [
    "function conformal_torus(sizerect::Tuple{Int, Int})\n",
    "    #conformal torus parameterization http://archive.bridgesmathart.org/2011/bridges2011-593.pdf\n",
    "    r, c = sizerect\n",
    "    a = c/r\n",
    "    ul = collect(range(-a*pi, a*pi, c))\n",
    "    vl = collect(range(-pi, pi, r))\n",
    "    u = ones(size(vl))*ul'\n",
    "    v = vl*ones(size(ul))'\n",
    "    tp = @. sqrt(a^2+1)-cos(v)\n",
    "    x = @. a*cos(u/a)/tp\n",
    "    y = @. a*sin(u/a)/tp\n",
    "    z = @. sin(v)/tp;\n",
    "    return x, y, z \n",
    "end"
   ]
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  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2b3c9101",
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   "source": [
    "img=load(\"Patterns/tilling.jpg\")\n",
    "r, c=size(img)\n",
    "rgbm = permutedims(channelview(img), (2,3,1))\n",
    "surfcolor=rgbm[:, :, 1];"
   ]
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  {
   "cell_type": "code",
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   "source": [
    "fig1=Plot(heatmap(z=surfcolor, colorscale=colors.deep, showscale=false),\n",
    "          Layout(width=div(c,2), height=div(r, 2), xaxis_visible=false, yaxis_visible=false,\n",
    "          margin=attr(t=2, r=2, b=2, l=2)))\n"
   ]
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  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b566b550",
   "metadata": {},
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    "x, y, z = conformal_torus(size(img));\n",
    "\n",
    "fig2 = Plot(surface(x=x, y=y, z=z, colorscale=colors.deep, surfacecolor=surfcolor, showscale=false), \n",
    "    Layout(width=700, height=450, margin=attr(t=2, r=2, b=2, l=2),\n",
    "        scene=attr(xaxis_visible=false, yaxis_visible=false, zaxis_visible=false)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "765e345f",
   "metadata": {},
   "source": [
    "![pl tilling](https://github.com/empet/Datasets/blob/master/Images/tesselationtoctorus.png?raw=true)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e5719840",
   "metadata": {},
   "source": [
    "![conformal torus](https://github.com/empet/Datasets/blob/master/Images/conformal_torus.png?raw=true)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7bc6f159",
   "metadata": {},
   "outputs": [],
   "source": []
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