sets/minokowskiAdd.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Initial implementation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×2 Array{Float64,2}:\n",
       " 0.0   0.0\n",
       " 1.0   1.0\n",
       " 1.0  -1.0"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = [1.0 0.0; 0.0 1.0; 0.0 -1.0]\n",
    "B = [0.0 0.0; 1.0 1.0; 1.0 -1.0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "minkowskiAdd (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function minkowskiAdd(A, B)\n",
    "    res = zeros((size(A)[1]*size(B)[1], 2))\n",
    "    n = 1\n",
    "    for i = 1:size(A)[1]  # comment: you can use 1:size(A, 1) here\n",
    "        for j = 1:size(B)[1] # comment: you can use 1:size(B, 1) here\n",
    "            res[n,1] = A[i,1]+B[j,1]\n",
    "            res[n,2] = A[i,2]+B[j,2]\n",
    "            n += 1\n",
    "        end\n",
    "    end\n",
    "    return res\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9×2 Array{Float64,2}:\n",
       " 1.0   0.0\n",
       " 2.0   1.0\n",
       " 2.0  -1.0\n",
       " 0.0   1.0\n",
       " 1.0   2.0\n",
       " 1.0   0.0\n",
       " 0.0  -1.0\n",
       " 1.0   0.0\n",
       " 1.0  -2.0"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minkowskiAdd(A, B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using LazySets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LazySets"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In LazySets there is the `VPolygon` type which can be used to build a polygon in vertex representation.\n",
    "For example, take the matrices $A$ and $B$ from the initial implementation, where each *row* corresponds to one\n",
    "vertex of a polygon."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below we use the programming construct that is called \"list comprehension\" to build the polygon:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "VPolygon{Float64}(Array{Float64,1}[[0.0, -1.0], [1.0, 0.0], [0.0, 1.0]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P = VPolygon([A[i, :] for i in 1:size(A, 1)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We do the same thing for `B`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "VPolygon{Float64}(Array{Float64,1}[[0.0, 0.0], [1.0, -1.0], [1.0, 1.0]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q = VPolygon([B[i, :] for i in 1:size(A, 1)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can plot these two:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "\n",
    "plot(P)\n",
    "\n",
    "plot!(Q) # use the bang operator (!) to put the new plot in the same graph as the previous one"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We know that the minkowski sum of these two polygons is the set defined as\n",
    "\n",
    "$$\n",
    " S = \\{ p+q : p \\in P, q \\in Q\\}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The idea in the function `minkowskiAdd` of the original implementation does precisely this calculation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9×2 Array{Float64,2}:\n",
       " 1.0   0.0\n",
       " 2.0   1.0\n",
       " 2.0  -1.0\n",
       " 0.0   1.0\n",
       " 1.0   2.0\n",
       " 1.0   0.0\n",
       " 0.0  -1.0\n",
       " 1.0   0.0\n",
       " 1.0  -2.0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = minkowskiAdd(A, B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us bring that matrix `C` into a polygon:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "VPolygon{Float64}(Array{Float64,1}[[0.0, -1.0], [1.0, -2.0], [2.0, -1.0], [2.0, 1.0], [1.0, 2.0], [0.0, 1.0]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R = VPolygon([C[i, :] for i in 1:size(C, 1)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What happened here? It only has six vertices, instead of nine! Some vertices have misteriously \"disappeared\". Actually, the constructor of `VPolygon` is \"smart\" in the sense that it removes the redundant vertices automatically."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can turn this option off to see:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "VPolygon{Float64}(Array{Float64,1}[[1.0, 0.0], [2.0, 1.0], [2.0, -1.0], [0.0, 1.0], [1.0, 2.0], [1.0, 0.0], [0.0, -1.0], [1.0, 0.0], [1.0, -2.0]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_redundant = VPolygon([C[i, :] for i in 1:size(C, 1)], apply_convex_hull=false)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can imagine, it is convenient that this option is `true` by default, since there is no interest in keeping redundant vertices, in general.\n",
    "\n",
    "Then, it would be nice that our new concrete Minkowski sum function also removes redundant vertices by default. We consider this next."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One can always remove the repeated vertices using the convex hull function. There exist many cool algorithms for taking the convex hull operation.\n",
    "\n",
    "In LazySets we have an algorithm for the convex hull in two dimensions that works quite well and is called\n",
    "Andrew's monotone chain algorithm (see the [wikipedia page](https://en.wikipedia.org/wiki/Convex_hull_algorithms) for more information on convex hull algorithms).\n",
    "\n",
    "For now we just want to modify `minkowskiAdd` such that it eliminates redundant vertices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "minkowskiAdd_v2 (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function minkowskiAdd_v2(A, B)\n",
    "    res = zeros((size(A)[1]*size(B)[1], 2))\n",
    "    n = 1\n",
    "    for i = 1:size(A)[1]\n",
    "        for j = 1:size(B)[1]\n",
    "            res[n,1] = A[i,1]+B[j,1]\n",
    "            res[n,2] = A[i,2]+B[j,2]\n",
    "            n += 1\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    # eliminate redundant vertices here\n",
    "    res = convex_hull([res[i, :] for i in 1:size(C, 1)])\n",
    "    \n",
    "    # a rather advanced technical detail: now res is a list of vertices (vector of vectors)\n",
    "    # you can transform it to a matrix as before with the following command\n",
    "    # res = Matrix(hcat(C...)')\n",
    "    # since we anyways need to pass the vector of vectors to the constructor of VPolygon,\n",
    "    # i wont do this here\n",
    "\n",
    "    # return the new res\n",
    "    return res\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try it!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6-element Array{Array{Float64,1},1}:\n",
       " [0.0, -1.0]\n",
       " [1.0, -2.0]\n",
       " [2.0, -1.0]\n",
       " [2.0, 1.0] \n",
       " [1.0, 2.0] \n",
       " [0.0, 1.0] "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = minkowskiAdd_v2(A, B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It has removed the redundant vertices. Now we would like to plot the result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
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     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "R = VPolygon(C)\n",
    "\n",
    "plot!(R)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And that's the Minkowski sum!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us compare to the *lazy* Minkowski sum available in LazySets..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
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       "<polyline clip-path=\"url(#clip4602)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:0.5; fill:none\" points=\"\n",
       "  891.86,775.359 1235.31,1118.81 1235.31,431.909 891.86,775.359 \n",
       "  \"/>\n",
       "<polygon clip-path=\"url(#clip4602)\" points=\"\n",
       "891.86,1118.81 1235.31,1462.26 1578.76,1118.81 1578.76,431.909 1235.31,88.4582 891.86,431.909 891.86,1118.81 891.86,1118.81 \n",
       "  \" fill=\"#3da44d\" fill-rule=\"evenodd\" fill-opacity=\"0.5\"/>\n",
       "<polyline clip-path=\"url(#clip4602)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:0.5; fill:none\" points=\"\n",
       "  891.86,1118.81 1235.31,1462.26 1578.76,1118.81 1578.76,431.909 1235.31,88.4582 891.86,431.909 891.86,1118.81 \n",
       "  \"/>\n",
       "<polygon clip-path=\"url(#clip4602)\" points=\"\n",
       "891.86,1118.81 1235.31,1462.26 1578.76,1118.81 1578.76,431.909 1235.31,88.4582 891.86,431.909 891.86,1118.81 891.86,1118.81 \n",
       "  \" fill=\"#c271d2\" fill-rule=\"evenodd\" fill-opacity=\"0.2\"/>\n",
       "<polyline clip-path=\"url(#clip4602)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:0.2; fill:none\" points=\"\n",
       "  891.86,1118.81 1235.31,1462.26 1578.76,1118.81 1578.76,431.909 1235.31,88.4582 891.86,431.909 891.86,1118.81 \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot!(P ⊕ Q, alpha=.2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "They seem to match :) Very good."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Next step"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The next step is to make a new function `minkowski_sum` that takes as input two `VPolygon` and computes its concrete Minkowski sum. This is *not* implemented in LazySets and it is the purpose of the issue [LazySets#666](https://github.com/JuliaReach/LazySets.jl/issues/666)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "ename": "MethodError",
     "evalue": "MethodError: no method matching minkowski_sum(::VPolygon{Float64}, ::VPolygon{Float64})",
     "output_type": "error",
     "traceback": [
      "MethodError: no method matching minkowski_sum(::VPolygon{Float64}, ::VPolygon{Float64})",
      "",
      "Stacktrace:",
      " [1] top-level scope at In[15]:1"
     ]
    }
   ],
   "source": [
    "minkowski_sum(P, Q)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Tip:* To write a function whose name already exists in a package (this is called *extending* a function) you have to do the following in Julia:\n",
    "\n",
    "```julia\n",
    "import LazySets: minkowski_sum\n",
    "\n",
    "function minkowski_sum(P::VPolygon, Q::VPolygon)\n",
    "    # add you new cool functionality here\n",
    "    # . . .\n",
    "end\n",
    "```"
   ]
  }
 ],
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