{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Voronoi Diagrams\n",
    "\n",
    "Voronoi diagrams are constructed from their sites (given in homogeneous coordinates).\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "$VD = new VoronoiPolyhedron(SITES=>[[1,1,1],[1,0,1],[1,-1,1],[1,1,-1],[1,0,-1],[1,-1,-1]]);\n",
    "$VD->VISUAL_VORONOI;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "Actually, via lifting to the standard paraboloid, Voronoi diagrams are derived from `Polytope`.  That's why they have `VERTICES`, `FACETS`, and such.\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2 -2 -2 1\n",
       "1 0 -2 1\n",
       "2 2 -2 1\n",
       "2 -2 2 1\n",
       "1 0 2 1\n",
       "2 2 2 1\n",
       "1 0 0 0\n"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print $VD->FACETS;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0 0 1 2\n",
       "0 1 0 2\n",
       "1 1/2 0 -1\n",
       "0 -1 0 2\n",
       "0 0 -1 2\n",
       "1 -1/2 0 -1\n"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<details><summary><pre style=\"display:inline\"><small>Click here for additional output</small></pre></summary>\n",
       "<pre>\n",
       "polymake: used package cdd\n",
       "  cddlib\n",
       "  Implementation of the double description method of Motzkin et al.\n",
       "  Copyright by Komei Fukuda.\n",
       "  http://www-oldurls.inf.ethz.ch/personal/fukudak/cdd_home/\n",
       "\n",
       "</pre>\n",
       "</details>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print $VD->VERTICES;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  }
 ],
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   "codemirror_mode": "perl",
   "file_extension": ".pl",
   "mimetype": "text/x-polymake",
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