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   "source": [
    "# Tutorial A11"
   ]
  },
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   "source": [
    "__1__ Write a function that takes three vectors $\\mathbf{a}$, $\\mathbf{b}$, $\\mathbf{c}$ in 2D (NumPy arrays) and checks whether they form a triangle, i.e. if $a + b = \\pm c$."
   ]
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   "source": [
    "__2__ Consider the vector $\\mathbf{v} = \\begin{pmatrix}0& 0& 1\\end{pmatrix}^\\mathrm{T}$ and the rotation matrix"
   ]
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   "source": [
    "\\begin{equation*}\n",
    "\\mathbf{R} =  \\begin{pmatrix} 1 & 0 & 0 \\\\\n",
    "0 & 0 & -1\\\\\n",
    "0 & 1 & 0\n",
    "\\end{pmatrix}\n",
    "\\end{equation*}"
   ]
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   "source": [
    "  - Rotate $\\mathbf{v}$ by $\\mathbf{R}$ to obtain $\\mathbf{v}^\\prime$"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "  - Compute the scalar product of $\\mathbf{v}$ and $\\mathbf{v}^\\prime$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "  - By which angle did we rotate along which axis?"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__3__ Consider the second Laplace spherical harmonic\n",
    "\n",
    "$$Y_1^{-1}(\\theta, \\phi) = \\frac{1}{2}\\sqrt{\\frac{3}{2\\pi}}\\sin\\theta\\mathrm{e}^{-i\\phi}$$\n",
    "\n",
    "  - Compute $Y$ on a grid of 10 points per dimension ($0 \\leq \\theta \\leq \\pi$, $0 \\leq \\phi < 2\\pi$)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__4__ Given 1000 randomly places points in 2D, compute all the pairwise distances according to the formula below (see example below on how to generate random numbers).\n",
    "\n",
    "$d_{ij} = \\sqrt{(x_i-x_j)^2+(y_i-y_j)^2}$ with  $i, j \\in (1,2,...,1000)$\n",
    "\n",
    "Use three different Methods for this and compare the times (see example below on how to stop the time in Python):\n",
    "\n",
    " - __a)__ A naive double for loop \n",
    " - __Advanced b)__ A vectorized version using meshgrids\n",
    " - __Advanced c)__ Outer subtraction\n",
    " - __Advanced d)__ The `cdist` function from the `scipy` module (`scipy.spatial.distance.cdist`)\n",
    "\n",
    "How many distances are larger than 0.5?"
   ]
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   "source": [
    "# How to generate random numbers in NumPy\n",
    "import numpy as np\n",
    "\n",
    "x = np.random.random(1000)\n",
    "# Gives you 1000 numbers in the half-open interval [0., 1.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Doing ...\n",
      "... stuff ...\n",
      "Total execution time: 2.0033 seconds\n"
     ]
    }
   ],
   "source": [
    "# Example on how to measure execution times\n",
    "\n",
    "import time\n",
    "# Import the time module from the standard library\n",
    "\n",
    "t0 = time.time()\n",
    "# Store the current system time\n",
    "\n",
    "print(\"Doing ...\")\n",
    "time.sleep(1)\n",
    "# do something (or do nothing)\n",
    "\n",
    "print(\"... stuff ...\")\n",
    "time.sleep(1)\n",
    "\n",
    "t = time.time() - t0\n",
    "# Get the current system time again and compute the ellapsed time\n",
    "\n",
    "print(f\"Total execution time: {t:.4f} seconds\")"
   ]
  }
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