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    "This notebook is part of the orix documentation https://orix.readthedocs.io. Links to the documentation won’t work from the notebook."
   ]
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   "source": [
    "## Visualizing point groups\n",
    "\n",
    "This tutorial shows point group symmetry operations $s$ in the stereographic projection.\n",
    "\n",
    "Vectors located on the upper (`z >= 0`) hemisphere are displayed as points (`o`), whereas vectors on the lower hemisphere are reprojected onto the upper hemisphere and shown as crosses (`+`) by default.\n",
    "For more information about plot formatting and visualization, see [Vector3d.scatter()](../reference/generated/orix.vector.Vector3d.scatter.rst).\n",
    "\n",
    "Further explanation of these figures is provided at http://xrayweb.chem.ou.edu/notes/symmetry.html#point."
   ]
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  {
   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "%matplotlib inline\n",
    "from matplotlib import pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "from orix import plot\n",
    "from orix.quaternion import Rotation, symmetry\n",
    "from orix.vector import Vector3d\n",
    "\n",
    "\n",
    "plt.rcParams.update({\"font.size\": 15})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For example, the $S = O$ or $432$ point group:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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    "nbsphinx-thumbnail": {
     "tooltip": "Point group symmetry operations visualized in the stereographic projection"
    },
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     "nbsphinx-thumbnail"
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   "outputs": [],
   "source": [
    "symmetry.O.plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$(\\mathbf{e_1}, \\mathbf{e_2}, \\mathbf{e_3})$ are the unit vectors of the standard Cartesian (orthonormal) reference frame (see the [crystal reference frame tutorial](../tutorials/crystal_reference_frame.ipynb) for more details).\n",
    "\n",
    "The stereographic projection of all point groups is shown below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# fmt: off\n",
    "schoenflies = [\n",
    "    \"C1\", \"Ci\",                                          # triclinic\n",
    "    \"C2x\", \"C2y\", \"C2z\", \"Csx\", \"Csy\", \"Csz\", \"C2h\",     # monoclinic\n",
    "    \"D2\", \"C2v\", \"D2h\",                                  # orthorhombic\n",
    "    \"C4\", \"S4\", \"C4h\", \"D4\", \"C4v\", \"D2d\", \"D4h\",        # tetragonal\n",
    "    \"C3\", \"S6\", \"D3x\", \"D3y\", \"D3\", \"C3v\", \"D3d\", \"C6\",  # trigonal\n",
    "    \"C3h\", \"C6h\", \"D6\", \"C6v\", \"D3h\", \"D6h\",             # hexagonal\n",
    "    \"T\", \"Th\", \"O\", \"Td\", \"Oh\",                          # cubic\n",
    "]\n",
    "# fmt: on\n",
    "\n",
    "assert len(symmetry._groups) == len(schoenflies)\n",
    "\n",
    "schoenflies = [\n",
    "    S for S in schoenflies if not (S.endswith(\"x\") or S.endswith(\"y\"))\n",
    "]\n",
    "\n",
    "assert len(schoenflies) == 32\n",
    "\n",
    "R = Rotation.from_axes_angles((-1, 8, 1), 65, degrees=True)\n",
    "\n",
    "fig, ax = plt.subplots(\n",
    "    nrows=8,\n",
    "    ncols=4,\n",
    "    figsize=(10, 20),\n",
    "    subplot_kw={\"projection\": \"stereographic\"},\n",
    "    layout=\"tight\",\n",
    ")\n",
    "ax = ax.ravel()\n",
    "\n",
    "for i, S in enumerate(schoenflies):\n",
    "    Si = getattr(symmetry, S)\n",
    "\n",
    "    R_Si = Si.outer(R)\n",
    "    v = R_Si * Vector3d.zvector()\n",
    "\n",
    "    # reflection in the projection plane (x-y) is performed internally in\n",
    "    # Symmetry.plot() or when using the `reproject=True` argument for\n",
    "    # Vector3d.scatter()\n",
    "    v_reproject = Vector3d(v.data.copy())\n",
    "    v_reproject.z *= -1\n",
    "\n",
    "    # the Symmetry marker formatting for vectors on the upper and lower\n",
    "    # hemisphere can be set using `kwargs` and `reproject_scatter_kwargs`,\n",
    "    # respectively, for Symmetry.plot()\n",
    "\n",
    "    # vectors on the upper hemisphere are shown as open circles\n",
    "    ax[i].scatter(v, marker=\"o\", c=\"None\", ec=\"C0\", s=150)\n",
    "    # vectors on the lower hemisphere are reprojected onto the upper\n",
    "    # hemisphere and shown as crosses\n",
    "    ax[i].scatter(v_reproject, marker=\"+\", c=\"C0\", s=150)\n",
    "\n",
    "    ax[i].set_title(f\"${S}$ $({Si.name})$\")\n",
    "    ax[i].set_labels(\"$e_1$\", \"$e_2$\", None)"
   ]
  }
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