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   "source": [
    "##  SYMBOLIC  ROTATIONS in 3D with sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "vy,vy,vz,theta,c,s,V = sp.symbols('vy vy vz theta c s V')\n",
    "phi = sp.symbols('phi')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Rx(theta,V):\n",
    "    \"\"\"\n",
    "    Rotation of a 3d vector V\n",
    "    of an angle theta around\n",
    "    the x-axis\n",
    "    \"\"\"\n",
    "    c = sp.cos(theta)\n",
    "    s = sp.sin(theta)\n",
    "    R = sp.Matrix(\n",
    "        [[1, 0, 0], \n",
    "         [0, c, -s], \n",
    "         [0, s, c]]\n",
    "    )\n",
    "    return R*V\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[                           1],\n",
       "[-3*sin(theta) + 2*cos(theta)],\n",
       "[ 2*sin(theta) + 3*cos(theta)]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V = sp.Matrix([[1],[2],[3]])\n",
    "Rx(theta,V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Ry(theta,V):\n",
    "    \"\"\"\n",
    "    Rotation of a 3d vector V\n",
    "    of an angle theta around\n",
    "    the y-axis\n",
    "    \"\"\"\n",
    "    c = sp.cos(theta)\n",
    "    s = sp.sin(theta)\n",
    "    R = sp.Matrix(\n",
    "        [[c, 0, s],\n",
    "         [0, 1, 0],\n",
    "         [-s, 0, c]]\n",
    "    )\n",
    "    return R*V\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Rz(theta,V):\n",
    "    \"\"\"\n",
    "    Rotation of a 3d vector V\n",
    "    of an angle theta around\n",
    "    the z-axis\n",
    "    \"\"\"\n",
    "    c = sp.cos(theta)\n",
    "    s = sp.sin(theta)\n",
    "    R = sp.Matrix(\n",
    "        [[c, -s, 0], \n",
    "         [s, c, 0],\n",
    "         [0, 0, 1]]\n",
    "    )\n",
    "    return R*V\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[ -sqrt(2)/2],\n",
       "[3*sqrt(2)/2],\n",
       "[          3]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rz(sp.pi/4,V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def R(theta,phi,V):\n",
    "    \"\"\"\n",
    "    Rotation of a 3d vector V\n",
    "    of an angle theta around\n",
    "    the y-axis followed by a rotation\n",
    "    of phi around the z-axis\n",
    "    \"\"\"\n",
    "    return Rz(phi,Ry(theta,V))\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Examples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[cos(phi)*cos(theta)],\n",
       "[sin(phi)*cos(theta)],\n",
       "[        -sin(theta)]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V = sp.Matrix([[1],[0],[0]])\n",
    "R(theta,phi,V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[sin(theta)*cos(phi)],\n",
       "[sin(phi)*sin(theta)],\n",
       "[         cos(theta)]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V = sp.Matrix([[0],[0],[1]])\n",
    "R(theta,phi,V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[I*sin(phi) + cos(phi)*cos(theta)],\n",
       "[sin(phi)*cos(theta) - I*cos(phi)],\n",
       "[                     -sin(theta)]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V = sp.Matrix([[1],[-sp.I],[0]])\n",
    "R(theta,phi,V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[ I*sin(phi) - cos(phi)*cos(theta)],\n",
       "[-sin(phi)*cos(theta) - I*cos(phi)],\n",
       "[                       sin(theta)]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V = sp.Matrix([[-1],[-sp.I],[0]])\n",
    "R(theta,phi,V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[sin(theta)*cos(phi)],\n",
       "[sin(phi)*sin(theta)],\n",
       "[         cos(theta)]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V = sp.Matrix([[0],[0],[1]])\n",
    "R(theta,phi,V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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