Plotly-Slice-in-volumetric-data.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Slice in volumetric data, via Plotly ##"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A volume included in a parallelepiped is described by the values of  a scalar field, $f(x,y,z)$, with $x\\in[a,b]$, $y\\in [c,d]$, $z\\in[e,f]$.\n",
    "A slice in this volume is visualized by coloring the surface of the slice, according to the values of the function f, restricted to that surface.\n",
    "\n",
    "In order to plot a planar or a nonlinear slice of equation z=s(x,y) one proceeds as follows:\n",
    "\n",
    "- define a meshgrid in x,y;\n",
    "- evaluate z=s(x,y)\n",
    "- define an instance of the Plotly Surface class, that represents the surface z=s(x,y)\n",
    "- this surface is colored according to the values,  f(x,y,z), at its points. More precisely, the normalized values of the function f  are mapped to a colormap/colorscale."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With obvious modications we get slices of equation $x=s(y,z), y=s(z,x)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import plotly.graph_objects as go\n",
    "from IPython"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define a function that returns a slice as a Plotly Surface:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_the_slice(x,y,z, surfacecolor):\n",
    "    return go.Surface(x=x,\n",
    "                      y=y,\n",
    "                      z=z,\n",
    "                      surfacecolor=surfacecolor,\n",
    "                      coloraxis='coloraxis')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_lims_colors(surfacecolor):# color limits for a slice\n",
    "    return np.min(surfacecolor), np.max(surfacecolor)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us plot the slices z=0 and y=-0.5 in the volume defined by:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "scalar_f = lambda x,y,z: x*np.exp(-x**2-y**2-z**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.linspace(-2,2, 50)\n",
    "y = np.linspace(-2,2, 50)\n",
    "x, y = np.meshgrid(x,y)\n",
    "z = np.zeros(x.shape)\n",
    "surfcolor_z = scalar_f(x,y,z)\n",
    "sminz, smaxz = get_lims_colors(surfcolor_z)\n",
    "\n",
    "slice_z = get_the_slice(x, y, z, surfcolor_z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.linspace(-2,2, 50)\n",
    "z = np.linspace(-2,2, 50)\n",
    "x, z = np.meshgrid(x,y)\n",
    "y = -0.5 * np.ones(x.shape)\n",
    "surfcolor_y = scalar_f(x,y,z)\n",
    "sminy, smaxy = get_lims_colors(surfcolor_y)\n",
    "vmin = min([sminz, sminy])\n",
    "vmax = max([smaxz, smaxy])\n",
    "slice_y = get_the_slice(x, y, z, surfcolor_y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to be able to compare  the two slices, we choose a unique interval of values to be mapped to the colorscale:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def colorax(vmin, vmax):\n",
    "    return dict(cmin=vmin,\n",
    "                cmax=vmax)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig1 = go.Figure(data=[slice_z, slice_y])\n",
    "fig1.update_layout(\n",
    "         title_text='Slices in volumetric data', \n",
    "         title_x=0.5,\n",
    "         width=700,\n",
    "         height=700,\n",
    "         scene_zaxis_range=[-2,2], \n",
    "         coloraxis=dict(colorscale='BrBG',\n",
    "                        colorbar_thickness=25,\n",
    "                        colorbar_len=0.75,\n",
    "                        **colorax(vmin, vmax)))            \n",
    "      \n",
    "#fig1.show()  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "\n",
       "        <iframe\n",
       "            width=\"700\"\n",
       "            height=\"700\"\n",
       "            src=\"https://chart-studio.plotly.com/~empet/13862\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.IFrame at 0x23da6904400>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import IFrame    \n",
    "IFrame('https://chart-studio.plotly.com/~empet/13862', width=700, height=700)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Oblique slice in volumetric data ###"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As an example we plot comparatively two slices: a slice through $z=0$ and an oblique planar slice, that is defined by rotating the plane z=0 by $\\alpha=\\pi/4$, about Oy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rotating the plane $z=c$ about Oy (from Oz towards Ox) with $\\alpha$ radians we get the plane of equation\n",
    "$z=c/\\cos(\\alpha)-x\\tan(\\alpha)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "alpha = np.pi/4\n",
    "x = np.linspace(-2, 2, 50)\n",
    "y = np.linspace(-2, 2, 50)\n",
    "x, y = np.meshgrid(x,y)\n",
    "z = -x * np.tan(alpha)\n",
    "\n",
    "surfcolor_obl = scalar_f(x,y,z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "smino, smaxo = get_lims_colors(surfcolor_obl)\n",
    "vmin = min([sminz, smino])\n",
    "vmax = max([smaxz, smaxo])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "slice_obl = get_the_slice(x,y,z, surfcolor_obl)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig2 = go.Figure(data=[slice_z, slice_obl], layout=fig1.layout)\n",
    "fig2.update_layout( coloraxis=colorax(vmin, vmax))\n",
    "\n",
    "#fig2.show() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"700\"\n",
       "            height=\"700\"\n",
       "            src=\"https://chart-studio.plotly.com/~empet/13864\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.IFrame at 0x23da6904518>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IFrame('https://chart-studio.plotly.com/~empet/13864', width=700, height=700)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.core.display import HTML\n",
    "def  css_styling():\n",
    "    styles = open(\"./custom.css\", \"r\").read()\n",
    "    return HTML(styles)\n",
    "css_styling()"
   ]
  }
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