transformations

transforms.ipynb

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  {
   "metadata": {},
   "cell_type": "markdown",
   "source": [
    "# transformations\n",
    "\n",
    "this page is great\n",
    "https://www.labri.fr/perso/nrougier/python-opengl/#transformationsb"
   ],
   "id": "f5862be824d9c6c4"
  },
  {
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     "end_time": "2025-06-27T17:52:48.575038Z",
     "start_time": "2025-06-27T17:52:48.565156Z"
    }
   },
   "cell_type": "code",
   "source": [
    "import sympy as sp\n",
    "\n",
    "# first, let's set up our transformation as a 4x4 matrix A\n",
    "# this matrix encodes our rotation and translation for transforming structures/maps\n",
    "a11, a12, a13, a14 = sp.symbols('a11 a12 a13 a14')\n",
    "a21, a22, a23, a24 = sp.symbols('a21 a22 a23 a24')\n",
    "a31, a32, a33, a34 = sp.symbols('a31 a32 a33 a34')\n",
    "a41, a42, a43, a44 = sp.symbols('a41 a42 a43 a44')\n",
    "\n",
    "A = sp.Matrix([\n",
    "    [a11, a12, a13, a14],\n",
    "    [a21, a22, a23, a24],\n",
    "    [a31, a32, a33, a34],\n",
    "    [a41, a42, a43, a44]\n",
    "])\n",
    "A"
   ],
   "id": "d32d3e1f0183f948",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[a11, a12, a13, a14],\n",
       "[a21, a22, a23, a24],\n",
       "[a31, a32, a33, a34],\n",
       "[a41, a42, a43, a44]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\\\a_{21} & a_{22} & a_{23} & a_{24}\\\\a_{31} & a_{32} & a_{33} & a_{34}\\\\a_{41} & a_{42} & a_{43} & a_{44}\\end{matrix}\\right]$"
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 4
  },
  {
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-06-27T17:57:54.551406Z",
     "start_time": "2025-06-27T17:57:54.537967Z"
    }
   },
   "cell_type": "code",
   "source": [
    "# now, let's set up a 3D vector representing an arbitrary point in our structure\n",
    "# (in homogenous coordinates)\n",
    "px, py, pz = sp.symbols('p_x p_y p_z')\n",
    "vector = sp.Matrix([\n",
    "    [px],\n",
    "    [py],\n",
    "    [pz],\n",
    "    [1]]\n",
    ")\n",
    "vector"
   ],
   "id": "88274056a8d4eb3f",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[p_x],\n",
       "[p_y],\n",
       "[p_z],\n",
       "[  1]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}p_{x}\\\\p_{y}\\\\p_{z}\\\\1\\end{matrix}\\right]$"
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 12
  },
  {
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-06-27T17:59:12.788424Z",
     "start_time": "2025-06-27T17:59:12.775669Z"
    }
   },
   "cell_type": "code",
   "source": [
    "# we transform the vector by left-multiplying it\n",
    "# this is what pymol is doing with the matrix internally\n",
    "# to transform atoms/voxels in map\n",
    "transformed_vector = A @ vector\n",
    "transformed_vector"
   ],
   "id": "9deaed022c969590",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[a11*p_x + a12*p_y + a13*p_z + a14],\n",
       "[a21*p_x + a22*p_y + a23*p_z + a24],\n",
       "[a31*p_x + a32*p_y + a33*p_z + a34],\n",
       "[a41*p_x + a42*p_y + a43*p_z + a44]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}a_{11} p_{x} + a_{12} p_{y} + a_{13} p_{z} + a_{14}\\\\a_{21} p_{x} + a_{22} p_{y} + a_{23} p_{z} + a_{24}\\\\a_{31} p_{x} + a_{32} p_{y} + a_{33} p_{z} + a_{34}\\\\a_{41} p_{x} + a_{42} p_{y} + a_{43} p_{z} + a_{44}\\end{matrix}\\right]$"
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 15
  },
  {
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-06-27T18:00:57.915502Z",
     "start_time": "2025-06-27T18:00:57.906498Z"
    }
   },
   "cell_type": "code",
   "source": [
    "# now, this is what the matrix in our uncropped MRC file looks like\n",
    "uncropped_matrix = sp.eye(4)\n",
    "uncropped_matrix"
   ],
   "id": "481c10860bba1d12",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[1, 0, 0, 0],\n",
       "[0, 1, 0, 0],\n",
       "[0, 0, 1, 0],\n",
       "[0, 0, 0, 1]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & 1 & 0\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$"
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 16
  },
  {
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-06-27T18:08:28.748643Z",
     "start_time": "2025-06-27T18:08:28.701398Z"
    }
   },
   "cell_type": "code",
   "source": [
    "# the identity matrix above represents no transformation\n",
    "# let's verify this by transforming it with our transformation matrix\n",
    "# notice that the result is the same as A\n",
    "#\n",
    "# this is why replacing the matrix on the uncropped map works\n",
    "# the existing transformation on the uncropped map was a no-op\n",
    "result = A @ uncropped_matrix\n",
    "result"
   ],
   "id": "8cd22199a23bf281",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[a11, a12, a13, a14],\n",
       "[a21, a22, a23, a24],\n",
       "[a31, a32, a33, a34],\n",
       "[a41, a42, a43, a44]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\\\a_{21} & a_{22} & a_{23} & a_{24}\\\\a_{31} & a_{32} & a_{33} & a_{34}\\\\a_{41} & a_{42} & a_{43} & a_{44}\\end{matrix}\\right]$"
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 20
  },
  {
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-06-27T18:08:33.397652Z",
     "start_time": "2025-06-27T18:08:33.392598Z"
    }
   },
   "cell_type": "code",
   "source": [
    "# this is what the cropped MRC file matrix looks like\n",
    "tx, ty, tz = sp.symbols('t_x t_y t_z')\n",
    "cropped_matrix = sp.Matrix(\n",
    "    [[1, 0, 0, tx],\n",
    "     [0, 1, 0, ty],\n",
    "     [0, 0, 1, tz],\n",
    "     [0, 0, 0,  1]]\n",
    ")\n",
    "cropped_matrix"
   ],
   "id": "157ab8ac452bcea7",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[1, 0, 0, t_x],\n",
       "[0, 1, 0, t_y],\n",
       "[0, 0, 1, t_z],\n",
       "[0, 0, 0,   1]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0 & t_{x}\\\\0 & 1 & 0 & t_{y}\\\\0 & 0 & 1 & t_{z}\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$"
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 21
  },
  {
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-06-27T18:11:22.802344Z",
     "start_time": "2025-06-27T18:11:22.779397Z"
    }
   },
   "cell_type": "code",
   "source": [
    "# if we apply the transformation correctly to the\n",
    "# cropped map matrix we will see that the result is not the same as\n",
    "# the initial transformation\n",
    "#\n",
    "# setting the matrix on the cropped map to your transformation matrix doesn't work\n",
    "# because this result != your initial transformation\n",
    "result = A @ cropped_matrix\n",
    "result"
   ],
   "id": "9ed05d2a7c50e0d6",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[a11, a12, a13, a11*t_x + a12*t_y + a13*t_z + a14],\n",
       "[a21, a22, a23, a21*t_x + a22*t_y + a23*t_z + a24],\n",
       "[a31, a32, a33, a31*t_x + a32*t_y + a33*t_z + a34],\n",
       "[a41, a42, a43, a41*t_x + a42*t_y + a43*t_z + a44]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}a_{11} & a_{12} & a_{13} & a_{11} t_{x} + a_{12} t_{y} + a_{13} t_{z} + a_{14}\\\\a_{21} & a_{22} & a_{23} & a_{21} t_{x} + a_{22} t_{y} + a_{23} t_{z} + a_{24}\\\\a_{31} & a_{32} & a_{33} & a_{31} t_{x} + a_{32} t_{y} + a_{33} t_{z} + a_{34}\\\\a_{41} & a_{42} & a_{43} & a_{41} t_{x} + a_{42} t_{y} + a_{43} t_{z} + a_{44}\\end{matrix}\\right]$"
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 23
  },
  {
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-06-27T18:16:45.603128Z",
     "start_time": "2025-06-27T18:16:45.598112Z"
    }
   },
   "cell_type": "code",
   "source": [
    "# so, how do we solve your problem?\n",
    "#\n",
    "# you need to\n",
    "# - calculate the correct transformation for the cropped map as above (A @ cropped_matrix)\n",
    "# - set this correct transform on the cropped map\n",
    "#\n",
    "# this chains the transformation with the existing one rather than replacing it"
   ],
   "id": "d13302d7b0bdc06a",
   "outputs": [],
   "execution_count": 26
  }
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