diff --git a/boom.ipynb b/boom.ipynb
index d63f2b5..e57865e 100644
--- a/boom.ipynb
+++ b/boom.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 255,
+   "execution_count": 43,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -54,6 +54,7 @@
     "class ArUcoMarker2D:\n",
     "    id: int\n",
     "    corners: Quad2D\n",
+    "    params: DiamondBoardParameter\n",
     "\n",
     "    @property\n",
     "    def np_corners(self):\n",
@@ -108,16 +109,16 @@
     "        params.border_length + params.chess_length * 2 + params.marker_border_length\n",
     "    )\n",
     "    return (\n",
-    "        ArUcoMarker2D(ids[0], tl_to_square(tl_0_x, tl_0_y, params.marker_leghth)),\n",
-    "        ArUcoMarker2D(ids[1], tl_to_square(tl_1_x, tl_1_y, params.marker_leghth)),\n",
-    "        ArUcoMarker2D(ids[2], tl_to_square(tl_2_x, tl_2_y, params.marker_leghth)),\n",
-    "        ArUcoMarker2D(ids[3], tl_to_square(tl_3_x, tl_3_y, params.marker_leghth)),\n",
+    "        ArUcoMarker2D(ids[0], tl_to_square(tl_0_x, tl_0_y, params.marker_leghth), params),\n",
+    "        ArUcoMarker2D(ids[1], tl_to_square(tl_1_x, tl_1_y, params.marker_leghth), params),\n",
+    "        ArUcoMarker2D(ids[2], tl_to_square(tl_2_x, tl_2_y, params.marker_leghth), params),\n",
+    "        ArUcoMarker2D(ids[3], tl_to_square(tl_3_x, tl_3_y, params.marker_leghth), params),\n",
     "    )\n"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 256,
+   "execution_count": 44,
    "metadata": {},
    "outputs": [
     {
@@ -138,7 +139,7 @@
     "fig = plt.figure()\n",
     "ax = fig.gca()\n",
     "ax.set_xlim((0, params.total_side_length))\n",
-    "ax.set_ylim((0, params.total_side_length))\n",
+    "ax.set_ylim((0, params.total_side_length)) # type: ignore\n",
     "ax.set_aspect(\"equal\")\n",
     "# set origin to top-left (from bottom-left)\n",
     "ax.invert_yaxis()\n",
@@ -154,7 +155,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 257,
+   "execution_count": 45,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -192,6 +193,11 @@
     "    \"\"\"\n",
     "    4x4 transformation matrix\n",
     "    \"\"\"\n",
+    "    _normal_vector: NDArray\n",
+    "    \"\"\"\n",
+    "    (2, 3)\n",
+    "    start (the center of the plane) and end (the normal vector), length 1\n",
+    "    \"\"\"\n",
     "\n",
     "    def __init__(self, items: Sequence[ArUcoMarker2D]):\n",
     "        self._ids = np.array([item.id for item in items])\n",
@@ -202,6 +208,11 @@
     "            [corners_2d, np.zeros((corners_2d.shape[0], 4, 1))], axis=-1\n",
     "        )\n",
     "        self._transform_matrix = np.eye(4)\n",
+    "        def center(items: Sequence[ArUcoMarker2D]):\n",
+    "            return np.mean([item.np_corners for item in items], axis=(0, 1))\n",
+    "        c = center(items)\n",
+    "        assert c.shape == (2,)\n",
+    "        self._normal_vector = np.array([(c[0], c[1], 0), (c[0], c[1], 0.1)])\n",
     "\n",
     "    @property\n",
     "    def ids(self):\n",
@@ -223,6 +234,13 @@
     "        return np.array(list(g()))\n",
     "    \n",
     "    @property\n",
+    "    def transformed_normal_vector(self):\n",
+    "        def g():\n",
+    "            for v in self._normal_vector:\n",
+    "                yield transform_point(self.transform_matrix, v)\n",
+    "        return np.array(list(g()))\n",
+    "    \n",
+    "    @property\n",
     "    def transformed_geometry_center(self):\n",
     "        return np.mean(self.transformed_corners, axis=(0, 1))\n",
     "    \n",
@@ -261,19 +279,19 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 258,
+   "execution_count": 46,
    "metadata": {},
    "outputs": [],
    "source": [
-    "plane = DiamondPlane3D(markers)\n",
+    "plane_a = DiamondPlane3D(markers)\n",
     "# plane.rotate(-np.pi/2, np.array([0, 1, 0]))\n",
     "# t_corners = plane.corners\n",
-    "t_corners = plane.transformed_corners"
+    "t_corners = plane_a.transformed_corners"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 259,
+   "execution_count": 47,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -298,9 +316,17 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 49,
    "metadata": {},
    "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[[0.2005     0.20049998 0.        ]\n",
+      " [0.2005     0.20049998 0.1       ]]\n"
+     ]
+    },
     {
      "data": {
       "application/vnd.plotly.v1+json": {
@@ -732,6 +758,26 @@
           0
          ]
         },
+        {
+         "marker": {
+          "size": 2
+         },
+         "mode": "markers+lines",
+         "name": "normal_a",
+         "type": "scatter3d",
+         "x": [
+          0.2004999965429306,
+          0.2004999965429306
+         ],
+         "y": [
+          0.2004999816417694,
+          0.2004999816417694
+         ],
+         "z": [
+          0,
+          0.1
+         ]
+        },
         {
          "marker": {
           "size": 1
@@ -1156,6 +1202,26 @@
           0.0350000128746033
          ]
         },
+        {
+         "marker": {
+          "size": 2
+         },
+         "mode": "markers+lines",
+         "name": "normal_b",
+         "type": "scatter3d",
+         "x": [
+          0.2004999965429306,
+          0.2004999965429306
+         ],
+         "y": [
+          1.2277083037334799e-17,
+          0.10000000000000002
+         ],
+         "z": [
+          0.21050001835823062,
+          0.21050001835823062
+         ]
+        },
         {
          "marker": {
           "size": 1
@@ -1579,6 +1645,26 @@
          "z": [
           0.03500001287460331
          ]
+        },
+        {
+         "marker": {
+          "size": 2
+         },
+         "mode": "markers+lines",
+         "name": "normal_c",
+         "type": "scatter3d",
+         "x": [
+          -1.2277084862200736e-17,
+          0.09999999999999999
+         ],
+         "y": [
+          0.21050000345706943,
+          0.21050000345706943
+         ],
+         "z": [
+          0.21050001835823062,
+          0.21050001835823065
+         ]
         }
        ],
        "layout": {
@@ -1586,19 +1672,19 @@
          "aspectmode": "cube",
          "xaxis": {
           "range": [
-           -0.1,
+           -0.41100000000000003,
            0.41100000000000003
           ]
          },
          "yaxis": {
           "range": [
-           -0.1,
+           -0.41100000000000003,
            0.41100000000000003
           ]
          },
          "zaxis": {
           "range": [
-           -0.1,
+           -0.41100000000000003,
            0.41100000000000003
           ]
          }
@@ -2434,10 +2520,9 @@
     "        y=corners[:, 1],\n",
     "        z=corners[:, 2],\n",
     "        mode='markers+lines',\n",
-    "        name=str(plane.ids[i]),\n",
+    "        name=str(plane_a.ids[i]),\n",
     "        marker=dict(size=1),\n",
     "    ))\n",
-    "    # also draw the corner id\n",
     "    for j, (x, y, z) in enumerate(corners):\n",
     "        fig.add_trace(go.Scatter3d(\n",
     "            x=[x],\n",
@@ -2448,6 +2533,18 @@
     "            textposition='middle center',\n",
     "            marker=dict(size=5),\n",
     "        ))\n",
+    "\n",
+    "print(plane_a.transformed_normal_vector)\n",
+    "# normal vector\n",
+    "fig.add_trace(go.Scatter3d(\n",
+    "    x=plane_a.transformed_normal_vector[:,0],\n",
+    "    y=plane_a.transformed_normal_vector[:,1],\n",
+    "    z=plane_a.transformed_normal_vector[:,2],\n",
+    "    mode='markers+lines',\n",
+    "    name='normal_a',\n",
+    "    marker=dict(size=2),\n",
+    "))\n",
+    "\n",
     "for i, corners in enumerate(t_corners_b):\n",
     "    fig.add_trace(go.Scatter3d(\n",
     "        x=corners[:, 0],\n",
@@ -2467,6 +2564,14 @@
     "            textposition='middle center',\n",
     "            marker=dict(size=5),\n",
     "        ))\n",
+    "fig.add_trace(go.Scatter3d(\n",
+    "    x=plane_b.transformed_normal_vector[:,0],\n",
+    "    y=plane_b.transformed_normal_vector[:,1],\n",
+    "    z=plane_b.transformed_normal_vector[:,2],\n",
+    "    mode='markers+lines',\n",
+    "    name='normal_b',\n",
+    "    marker=dict(size=2),\n",
+    "))\n",
     "\n",
     "for i, corners in enumerate(t_corners_c):\n",
     "    fig.add_trace(go.Scatter3d(\n",
@@ -2488,25 +2593,32 @@
     "            textposition='middle center',\n",
     "            marker=dict(size=5),\n",
     "        ))\n",
-    "                      \n",
+    "fig.add_trace(go.Scatter3d(\n",
+    "    x=plane_c.transformed_normal_vector[:,0],\n",
+    "    y=plane_c.transformed_normal_vector[:,1],\n",
+    "    z=plane_c.transformed_normal_vector[:,2],\n",
+    "    mode='markers+lines',\n",
+    "    name='normal_c',\n",
+    "    marker=dict(size=2),\n",
+    "))\n",
     "\n",
-    "# fig.update_layout(\n",
-    "#     scene=dict(\n",
-    "#         aspectmode='cube',\n",
-    "#         xaxis=dict(range=[-params.total_side_length, params.total_side_length]),\n",
-    "#         yaxis=dict(range=[-params.total_side_length, params.total_side_length]),\n",
-    "#         zaxis=dict(range=[-params.total_side_length, params.total_side_length]),\n",
-    "#         # zaxis=dict(range=[0, params.total_side_length]),\n",
-    "#     )\n",
-    "# )\n",
     "fig.update_layout(\n",
     "    scene=dict(\n",
     "        aspectmode='cube',\n",
-    "        xaxis=dict(range=[-0.1, params.total_side_length]),\n",
-    "        yaxis=dict(range=[-0.1, params.total_side_length]),\n",
-    "        zaxis=dict(range=[-0.1, params.total_side_length]),\n",
+    "        xaxis=dict(range=[-params.total_side_length, params.total_side_length]),\n",
+    "        yaxis=dict(range=[-params.total_side_length, params.total_side_length]),\n",
+    "        zaxis=dict(range=[-params.total_side_length, params.total_side_length]),\n",
+    "        # zaxis=dict(range=[0, params.total_side_length]),\n",
     "    )\n",
     ")\n",
+    "# fig.update_layout(\n",
+    "#     scene=dict(\n",
+    "#         aspectmode='cube',\n",
+    "#         xaxis=dict(range=[-0.1, params.total_side_length]),\n",
+    "#         yaxis=dict(range=[-0.1, params.total_side_length]),\n",
+    "#         zaxis=dict(range=[-0.1, params.total_side_length]),\n",
+    "#     )\n",
+    "# )\n",
     "fig.show()"
    ]
   }