{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Tetrahedral Mesh"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from lapy import TetMesh"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, instead of loading, we define a small tetrahedral mesh representing a cube with a center vertex and twelve tetrahedra."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# cube9 (cube with center node)\n",
    "points = [\n",
    "    [0, 0, 0],\n",
    "    [1, 0, 0],\n",
    "    [1, 1, 0],\n",
    "    [0, 1, 0],\n",
    "    [0, 0, 1],\n",
    "    [1, 0, 1],\n",
    "    [1, 1, 1],\n",
    "    [0, 1, 1],\n",
    "    [0.5, 0.5, 0.5],\n",
    "]\n",
    "tets = [\n",
    "    [0, 5, 8, 1],\n",
    "    [0, 4, 5, 8],\n",
    "    [2, 5, 6, 8],\n",
    "    [1, 5, 2, 8],\n",
    "    [6, 7, 3, 8],\n",
    "    [6, 3, 2, 8],\n",
    "    [0, 3, 4, 8],\n",
    "    [3, 7, 4, 8],\n",
    "    [0, 1, 2, 8],\n",
    "    [0, 2, 3, 8],\n",
    "    [4, 6, 5, 8],\n",
    "    [4, 7, 6, 8],\n",
    "]\n",
    "T = TetMesh(points, tets)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note, that we flipped the first tetrahedron vertex order on purpose (it should be 0,5,1,8) to test and correct orientation below."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can check if our tet mesh has free vertices (these are vertices that are not used in any tetrahedron)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T.has_free_vertices()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "No free vertices are found, so we cannot remove any and the attempt will keep all 9."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([0, 1, 2, 3, 4, 5, 6, 7, 8]), [])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T.rm_free_vertices_()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see next, if we have consistent orientations (this should fail)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Orientations are not uniform\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T.is_oriented()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some functions don't care about the orientation, for example the average edge length computation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0543647924813107"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T.avg_edge_length()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also get the boundary of the tet mesh as a triangle mesh."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Found 12 triangles on boundary.\n"
     ]
    }
   ],
   "source": [
    "BT = T.boundary_tria()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But also the boundary is not oriented consistently (triangle normals of neighboring triangles point in opposite directions)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BT.is_oriented()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's repeat those steps after correcting the orientation in the tet mesh."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Flipped 1 tetrahedra\n",
      "All tet orientations are correct\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T.orient_()\n",
    "T.is_oriented()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When we extract the boundary surface now, we see it is also consistently oriented."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Found 12 triangles on boundary.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BT = T.boundary_tria()\n",
    "BT.is_oriented()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Correct orientation is needed, e.g., to compute the volume of a surface mesh."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BT.volume()"
   ]
  }
 ],
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