module ConvexHull where

import qualified Data.Set as S (fromList, toAscList)
import qualified Data.List as L (init, foldl')

data Point a = Point
           { xCoord :: a
           , yCoord :: a
           }
           deriving(Eq, Ord)

data Vec a = Vec
         { xComp :: a
         , yComp :: a
         }

direction :: Num a => Point a -> Point a -> Vec a
direction (Point x1 y1) (Point x2 y2) = Vec (x2 - x1) (y2 - y1)

innerProd :: Num a => Vec a -> Vec a -> a
innerProd (Vec x1 y1) (Vec x2 y2) = x1*x2 + y1*y2

-- Check whether the three points make a convex angle
convexAngle :: (Ord a, Num a) => Point a -> Point a -> Point a -> Bool
convexAngle p1 p2 p3 = let dir1 = p1 `direction` p2
                           dir2 = p2 `direction` p3
                       in (dir1 `innerProd` dir2) > 0


removeNonConvexAngles :: (Ord a, Num a) => [Point a] -> [Point a]
removeNonConvexAngles ps@(p1:p2:p3:rest)
    | convexAngle p3 p2 p1 = ps
    | otherwise            = removeNonConvexAngles (p1:p3:rest)

-- if there are less than three points, there is nothing to do
removeNonConvexAngles ps = ps

convexHullHelper :: (Ord a, Num a) => [Point a] -> Point a -> [Point a]
convexHullHelper ps newPoint = removeNonConvexAngles (newPoint : ps)


-- Given a set of points, return the list of points (ordered clockwisely)
-- that form the vertices of the convex hull
convexHull :: (Ord a, Num a) => [Point a] -> [Point a]
convexHull points =
    let orderedPoints = S.toAscList (S.fromList points)
        upperConvexHull = L.foldl' convexHullHelper [] orderedPoints
        lowerConvexHull = foldr (flip convexHullHelper) [] orderedPoints
        -- Remove first and last element from the lower convex hull, because they are duplicates
        lowerConvexHull' = tail (L.init lowerConvexHull)

    in upperConvexHull ++ lowerConvexHull'