{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "{-# LANGUAGE TypeOperators #-}\n",
    "import Prelude ()\n",
    "import qualified Prelude\n",
    "import Data.Manifold.TreeCover\n",
    "import Data.Manifold.Web\n",
    "import Data.Random\n",
    "import Data.Random.Manifold\n",
    "import Data.Manifold.Types\n",
    "import Math.LinearMap.Category\n",
    "import Linear(V2(..))\n",
    "import Data.VectorSpace\n",
    "import Data.Semigroup\n",
    "import qualified Data.List.NonEmpty as NE\n",
    "import Data.Foldable (toList)\n",
    "import Control.Monad (replicateM)\n",
    "\n",
    "import Control.Category.Constrained.Prelude\n",
    "import Control.Arrow.Constrained\n",
    ":opt no-lint  -- lint gives bogus warnings with constrained-categories"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "From [dynamic-plot](http://hackage.haskell.org/package/dynamic-plot):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import Graphics.Dynamic.Plot.R2 -- hiding (plotWindow)\n",
    "-- plotWindow _ = return ()\n",
    "import Diagrams.Prelude ((^&), (&), r2, white, opacity, fc)\n",
    "\n",
    "plotPoints :: [ℝ²] -> DynamicPlottable\n",
    "plotPoints ps = plot [p|±|[0.01^&0, 0^&0.01] | p<-ps]\n",
    "\n",
    "viewShadeCombination :: [Shade' ℝ²] -> [DynamicPlottable]\n",
    "viewShadeCombination [] = []\n",
    "viewShadeCombination shs' = [ plot . sconcat $ ellipsoid<$>shs\n",
    "                            , plot (mixShade's shs) & tweakPrerendered (opacity 0.3 . fc white) ]\n",
    " where shs = NE.fromList shs'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GraphWindowSpecR2{lBound=-1.5, rBound=2.4999999999999996, bBound=-0.1666666666666666, tBound=1.1666666666666665, xResolution=640, yResolution=480}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "GraphWindowSpecR2{lBound=-1.5000000000000002, rBound=2.5, bBound=-0.16666666666666663, tBound=1.1666666666666665, xResolution=640, yResolution=480}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotWindow $ [plot $ viewShadeCombination [V2 0 y|±|[V2 0.5 0, V2 0 0.1], V2 y y|±|[V2 0.5 0,V2 0 0.1]]\n",
    "             | y <- [0.1,0.3 .. 1.5] ]\n",
    "          ++ (plot<$>[xInterval (-1,2), yInterval (0,1)])\n",
    "plotWindow $ [plot $ viewShadeCombination [V2 0 y|±|[V2 0.6 0,V2 0 0.1], V2 y y|±|[V2 0.33 0,V2 0 0.1]]\n",
    "             | y <- [0.1,0.3 .. 1.5] ]\n",
    "          ++ (plot<$>[xInterval (-1,2), yInterval (0,1)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Convex intersection of equal, cospectral ellipses](https://raw.githubusercontent.com/leftaroundabout/manifolds/master/manifolds/images/examples/ShadeCombinations/EqualCospecEllipseIntersections.png)\n",
    "![Convex intersection of nonsimilar ellipses](https://raw.githubusercontent.com/leftaroundabout/manifolds/master/manifolds/images/examples/ShadeCombinations/EllipseIntersections.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GraphWindowSpecR2{lBound=-1.5000000000000004, rBound=2.5000000000000004, bBound=-0.16666666666666669, tBound=1.166666666666667, xResolution=640, yResolution=480}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotWindow $ [plot $ viewShadeCombination [V2 0 y|±|[V2 0.5 0,V2 0 0.1], V2 0.5 (1.2*y)|±|[V2 0.5 0,V2 0 0.1]]\n",
    "             | y <- [0.1,0.3 .. 1.5] ]\n",
    "          ++ (plot<$>[xInterval (-1,2), yInterval (0,1)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Convex intersection of equal, sideways offset ellipses](https://raw.githubusercontent.com/leftaroundabout/manifolds/master/manifolds/images/examples/ShadeCombinations/EqualCospecOffsetEllipseIntersections.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GraphWindowSpecR2{lBound=-0.9166666666666667, rBound=2.416666666666666, bBound=-0.33333333333333326, tBound=2.333333333333333, xResolution=640, yResolution=480}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotWindow $ [plot $ viewShadeCombination [V2 0 0|±|[V2 0.5 (-0.5)], V2 2 0|±|[V2 0.3 0.4]]]\n",
    "          ++ (plot<$>[xInterval (-0.5,2), yInterval (0,2)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Convex intersection of two degenerate shades (infinite extension in some direction)](https://raw.githubusercontent.com/leftaroundabout/manifolds/master/manifolds/images/examples/ShadeCombinations/DifferentDegenerateIntersections.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GraphWindowSpecR2{lBound=-0.9166666666666667, rBound=2.416666666666666, bBound=-0.33333333333333326, tBound=2.3333333333333335, xResolution=640, yResolution=480}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotWindow $ [plot $ viewShadeCombination [V2 0 0|±|[V2 0.5 (-0.5), V2 1 1], V2 2 0|±|[V2 0.3 0.4]]]\n",
    "          ++ (plot<$>[xInterval (-0.5,2), yInterval (0,2)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Convex intersection of a proper shade with a degenerate one](https://raw.githubusercontent.com/leftaroundabout/manifolds/master/manifolds/images/examples/ShadeCombinations/SinglyDegenerateIntersections.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GraphWindowSpecR2{lBound=-1.0, rBound=6.999999999999999, bBound=-2.1666666666666665, tBound=3.166666666666666, xResolution=640, yResolution=480}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotWindow $ [ plot $ viewShadeCombination [V2 1 0|±|[V2 1 0, V2 0 3], V2 0 0|±|[V2 (-0.3) 0.4]]\n",
    "             , plot $ viewShadeCombination [V2 4 0|±|[V2 (-0.3) 0.4], V2 5 0|±|[V2 1 0, V2 0 3]] ]\n",
    "          ++ (plot<$>[xInterval (0,6), yInterval (-1.5,2.5)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GraphWindowSpecR2{lBound=-0.6666666666666664, rBound=4.666666666666668, bBound=-1.5, tBound=2.5, xResolution=640, yResolution=480}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotWindow $ viewShadeCombination [ V2 0   0|±|[V2 1 0,V2 0 1]\n",
    "                                  , V2 0.4 0|±|[V2 0.5 0,V2 0 0.5]\n",
    "                                  , V2 1   0|±|[V2 1 0,V2 0 1]\n",
    "                                  , V2 0.5 1|±|[V2 1 0,V2 0 1] ]\n",
    "          ++ viewShadeCombination [ V2 3   0|±|[V2 1 0,V2 0 1]\n",
    "                                  , V2 3.4 0|±|[V2 1.3 0,V2 0 0.4]\n",
    "                                  , V2 4   0|±|[V2 1 0,V2 0 1]\n",
    "                                  , V2 3.5 1|±|[V2 1 0,V2 0 1] ]\n",
    "          ++ (plot<$>[xInterval (0,4), yInterval (-1,2)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Convex intersection of ellipsoid sets](https://raw.githubusercontent.com/leftaroundabout/manifolds/master/manifolds/images/examples/ShadeCombinations/ConvexIntersections.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Shade' {_shade'Ctr = 0.0, _shade'Narrowness = spanNorm [0.5000000000000002]}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "convolveShade' (0|±|[1]) (0|±|[1]) :: Shade' ℝ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Shade' {_shade'Ctr = (0.0,0.0), _shade'Narrowness = spanNorm [(9.999999999983312e-3,0.0),(0.0,9.999999999983312e-3)]}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "convolveShade' ((0,0)|±|[(1,0),(0,1)]) ((0,0)|±|[(99,0),(0,99)]) :: Shade' (ℝ,ℝ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "convoTests :: [[Shade' (ℝ,ℝ)]]\n",
    "convoTests = [ [ (0,6) |±| [(4,0),(0,2)]\n",
    "               , (9,4) |±| [(4,0),(0,2)] ]\n",
    "             , [ (8,0) |±| [(4,0),(0,1)]\n",
    "               , (0,8) |±| [(1,0),(0,4)] ]\n",
    "             , [ (9,0) |±| [(2  ,2) ,(0.1,2.5)]\n",
    "               , (0,8) |±| [(0.5,-0.1),(1,3)] ]\n",
    "             , [ (8,0) |±| [(1  ,-0.7)]\n",
    "               , (0,8) |±| [(0.7,-0.2),(2.7 ,2  )] ]\n",
    "             , [ (2,6) |±| [(4,0),(0,4)]\n",
    "               , (9,0) |±| [(1,0),(0,0.1)] ] ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": []
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "forM_ convoTests $ \\wings -> do\n",
    "  let convo = convolveShade' (wings!!0) (wings!!1)\n",
    "  \n",
    "  testPts <- runRVar (forM wings $ \\sh@(Shade' cs es) ->\n",
    "    fmap (filter $ \\p -> occlusion sh p > exp(-1))\n",
    "     . replicateM 150 $ do\n",
    "        φ <- uniform 0 (2*pi)\n",
    "        r <- normal 1 0.1\n",
    "        let [v₁,v₂] = normSpanningSystem' es\n",
    "        return $ cs ^+^ (r*cos φ)*^v₁ ^+^ (r*sin φ)*^v₂\n",
    "   ) StdRandom  :: IO [[(ℝ,ℝ)]]\n",
    "\n",
    "  plotWindow $ [ plot [plot sh, plotPoints $ r2 <$> tps]\n",
    "               | (sh,tps) <- zip (convo:wings) ([a^+^b | a<-testPts!!0, b<-testPts!!1] : testPts)]\n",
    "            ++ (plot<$>[xInterval (0,16), yInterval (1,12)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "<a id='shadeConvolutions'></a>\n",
    "![Convolution of coaligned shades](https://raw.githubusercontent.com/leftaroundabout/manifolds/master/manifolds/images/examples/ShadeCombinations/2Dconvolution-coaligned.png)\n",
    "![Convolution of \"orthogonal\" shades](https://raw.githubusercontent.com/leftaroundabout/manifolds/master/manifolds/images/examples/ShadeCombinations/2Dconvolution-orthogonal.png)\n",
    "![Convolution of skewed shades](https://raw.githubusercontent.com/leftaroundabout/manifolds/master/manifolds/images/examples/ShadeCombinations/2Dconvolution-skewed.png)\n",
    "![Convolution of a compact shade and a degenerate one](https://raw.githubusercontent.com/leftaroundabout/manifolds/master/manifolds/images/examples/ShadeCombinations/2Dconvolution-degenerate.png)\n",
    "![Convolution of shades with very different shape](https://raw.githubusercontent.com/leftaroundabout/manifolds/master/manifolds/images/examples/ShadeCombinations/2Dconvolution-big+small.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GraphWindowSpecR2{lBound=-2.456011329583298, rBound=13.192079307083088, bBound=-0.9969665083072043, tBound=4.983644332615075, xResolution=640, yResolution=480}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import Data.Manifold.Riemannian\n",
    "\n",
    "interpolate₀, interpolate₁ :: Shade (ℝ,ℝ)\n",
    "interpolate₀ = (0,1):±[(0.5,0),(0,1)]\n",
    "interpolate₁ = (9,2):±[(2,1),(1,2)]\n",
    "\n",
    "Just interpolation = geodesicBetween interpolate₀ interpolate₁\n",
    "\n",
    "plotWindow [ plot interpolate₀\n",
    "           , plot [ plot $ interpolation (D¹ x)\n",
    "                  | x <- [-0.6, -0.2 .. 0.6] ]\n",
    "           , plot interpolate₁]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Interpolating between shades, with location and extend](https://raw.githubusercontent.com/leftaroundabout/manifolds/master/manifolds/images/examples/ShadeCombinations/interpolation_straight2skewed.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GraphWindowSpecR2{lBound=-2.4560113295832986, rBound=13.192079307083088, bBound=-0.9814239699997196, tBound=4.981423969999719, xResolution=640, yResolution=480}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interpolate₀, interpolate₁ :: Shade (ℝ,ℝ)\n",
    "interpolate₀ = (0,1):±[(0.5,-0.5)]\n",
    "interpolate₁ = (9,2):±[(2,1),(1,2)]\n",
    "\n",
    "Just interpolation = geodesicBetween interpolate₀ interpolate₁\n",
    "\n",
    "plotWindow [ plot interpolate₀\n",
    "           , plot [ plot $ interpolation (D¹ x)\n",
    "                  | x <- [-0.6, -0.2 .. 0.6] ]\n",
    "           , plot interpolate₁]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GraphWindowSpecR2{lBound=-1.3333333333333328, rBound=1.3333333333333328, bBound=-1.3333333333333328, tBound=1.3333333333333328, xResolution=640, yResolution=480}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interpolate₀', interpolate₁' :: Shade' (ℝ,ℝ)\n",
    "interpolate₀' = (-1,-1)|±|[(0.1,0),(0,0.2)]\n",
    "interpolate₁' = (1,1)  |±|[(0.1,0.2),(-0.3,0.3)]\n",
    "\n",
    "Just interpolation' = geodesicBetween interpolate₀' interpolate₁'\n",
    "\n",
    "plotWindow [ plot interpolate₀'\n",
    "           , plot [ plot $ interpolation' (D¹ x)\n",
    "                  | x <- [-0.6, -0.2 .. 0.6] ]\n",
    "           , plot interpolate₁']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Interpolating between co-shades, with location and extend](https://raw.githubusercontent.com/leftaroundabout/manifolds/master/manifolds/images/examples/ShadeCombinations/interpolation'_straight2skewed.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GraphWindowSpecR2{lBound=-1.333333333333333, rBound=1.333333333333333, bBound=-1.333333333333333, tBound=1.333333333333333, xResolution=640, yResolution=480}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "let pss = [[(-1)^&0, (-1)^&0.1, 0^&0, 0^&(0.1), 1^&0.2, 1^&ye :: ℝ²] | ye <- [0.3,0.31..]]\n",
    " in plotWindow [plotLatest [[plotPoints ps, plotLatest (pointsCover's ps :: [Shade' ℝ²])] | ps<-pss]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "![Convex hull of a set of points with an outlier](https://raw.githubusercontent.com/leftaroundabout/manifolds/master/manifolds/images/examples/ShadeCombinations/OutlierPoints-ConvexHull.gif)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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