module Shape
        ( Shape
        , area
        , square
        , circle
        , polygon
        )
where

data Shape = Rectangle Side Side
           | Ellipse Radius Radius
           | RtTriangle Side Side
           | Polygon [Vertex]
  deriving Show

type Radius = Float
type Side = Float
type Vertex = (Float, Float)

polygon :: [Vertex] -> Shape
polygon vs = Polygon vs

area :: Shape -> Float
area (Rectangle  s1 s2 ) = s1 * s2
area (RtTriangle s1 s2 ) = s1 * s2 / 2
area (Ellipse    r1 r2 ) = pi * r1 * r2
area (Polygon vs)
  | length vs <= 2 = 0
  | otherwise = polyArea (vs ++ [head vs])
  where polyArea :: [Vertex] -> Float
        polyArea (v1: v2: vs') = trapezoidArea v1 v2 + polyArea (v2:vs')
        polyArea _ = 0

trapezoidArea :: Vertex -> Vertex -> Float
trapezoidArea v1 v2 =
        let h      = fst v2 - fst v1
            aplusb = snd v2 + snd v1
        in  (aplusb / 2) * h

triArea :: Vertex -> Vertex -> Vertex -> Float
triArea v1 v2 v3 =
        let a = distBetween v1 v2
            b = distBetween v2 v3
            c = distBetween v3 v1
            s = 0.5 * (a + b + c)
        in  sqrt (s * (s - a) * (s - b) * (s - c))

distBetween :: Vertex -> Vertex -> Float
distBetween (x1, y1) (x2, y2) = sqrt ((x1 - x2) ^ 2 + (y1 - y2) ^ 2)

circle :: Radius -> Shape
circle r = Ellipse r r

convex :: Shape -> Bool
convex (Rectangle  a  b ) = True
convex (RtTriangle a  b ) = True
convex (Ellipse    r1 r2) = True
convex (Polygon [_, _, _]) = True
convex (Polygon (vfirst : vsecond : vthird : vs)) = False

crossProduct :: Vertex -> Vertex -> Vertex -> Float
crossProduct (x1, y1) (x2, y2) (x3, y3) = (x2 - x1) * (y3 - y2) - (y2 - y1) * (x3 - x2)

square :: Side -> Shape
square s = Rectangle s s

rectangle :: Side -> Side -> Shape
rectangle s1 s2 = Polygon [(x, y), (-x, y), (-x, -y), (x, -y)]
    where
        x = s1 / 2
        y = s2 / 2

regularPolygon :: Int -> Side -> Shape
regularPolygon n s =
        let
                angleinc = (pi * 2) / fromIntegral n
                radius   = (s * sin ((pi - angleinc) / 2)) / sin angleinc
                regularVerts 0 _ = []
                regularVerts n angle =
                        (radius * cos angle, radius * sin angle) : regularVerts
                                (n - 1)
                                (angle + angleinc)
        in
                Polygon (regularVerts n 0)

rtTriangle :: Side -> Side -> Shape
rtTriangle s1 s2 = Polygon [(0, 0), (s1, 0), (0, s2)]