{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeFamilies #-}

module FixedVector
where

import Data.List (intercalate)
import Data.Vector.Fixed as V
import Data.Vector.Fixed.Boxed

import Normed

-- | The Vn newtype simply wraps (Vector v a) so that we avoid
--   undecidable instances.
newtype Vn a = Vn a


instance (Show a, Vector v a) => Show (Vn (v a)) where
  -- | Display vectors as ordinary tuples. This is poor practice, but
  --   these results are primarily displayed interactively and
  --   convenience trumps correctness (said the guy who insists his
  --   vector lengths be statically checked at compile-time).
  --
  --   Examples:
  --
  --   >>> let v1 = make2d (1,2)
  --   >>> show v1
  --   (1,2)
  --
  show (Vn v1) =
    "(" ++ (intercalate "," element_strings) ++ ")"
    where
      v1l = toList v1
      element_strings = Prelude.map show v1l


-- | We would really like to say, "anything that is a vector of
--   equatable things is itself equatable." The 'Vn' class
--   allows us to express this without a GHC battle.
--
--   Examples:
--
--   >>> let v1 = make2d (1,2)
--   >>> let v2 = make2d (1,2)
--   >>> let v3 = make2d (3,4)
--   >>> v1 == v2
--   True
--   >>> v1 == v3
--   False
--
instance (Eq a, Vector v a, Vector v Bool) => Eq (Vn (v a)) where
  (Vn v1) == (Vn v2) = V.foldl (&&) True (V.zipWith (==) v1 v2)


-- | The use of 'Num' here is of course incorrect (otherwise, we
--   wouldn't have to throw errors). But it's really nice to be able
--   to use normal addition/subtraction.
instance (Num a, Vector v a) => Num (Vn (v a)) where
  -- | Componentwise addition.
  --
  --   Examples:
  --
  --   >>> let v1 = make2d (1,2)
  --   >>> let v2 = make2d (3,4)
  --   >>> v1 + v2
  --   (4,6)
  --
  (Vn v1) + (Vn v2) = Vn $ V.zipWith (+) v1 v2

  -- | Componentwise subtraction.
  --
  --   Examples:
  --
  --   >>> let v1 = make2d (1,2)
  --   >>> let v2 = make2d (3,4)
  --   >>> v1 - v2
  --   (-2,-2)
  --
  (Vn v1) - (Vn v2) = Vn $ V.zipWith (-) v1 v2

  -- | Create an n-vector whose components are all equal to the given
  --   integer. The result type must be specified since otherwise the
  --   length n would be unknown.
  --
  --   Examples:
  --
  --   >>> let v1 = fromInteger 17 :: Vn (Vec3 Int)
  --   (17,17,17)
  --
  fromInteger x = Vn $ V.replicate (fromInteger x)
  (*) = error "multiplication of vectors is undefined"
  abs = error "absolute value of vectors is undefined"
  signum = error "signum of vectors is undefined"

instance Functor Vn where
  fmap f (Vn v1) = Vn (f v1)

instance (RealFloat a, Ord a, Vector v a) => Normed (Vn (v a)) where
  -- | The infinity norm. We don't use V.maximum here because it
  --   relies on a type constraint that the vector be non-empty and I
  --   don't know how to pattern match it away.
  --
  --   Examples:
  --
  --   >>> let v1 = make3d (1,5,2)
  --   >>> norm_infty v1
  --   5
  --
  norm_infty (Vn v1) = fromRational $ toRational $ V.foldl max 0 v1

  -- | Generic p-norms. The usual norm in R^n is (norm_p 2).
  --
  --   Examples:
  --
  --   >>> let v1 = make2d (3,4)
  --   >>> norm_p 1 v1
  --   7.0
  --   >>> norm_p 2 v1
  --   5.0
  --
  norm_p p (Vn v1) =
    fromRational $ toRational $ root $ V.sum $ V.map (exponentiate . abs) v1
    where
      exponentiate = (** (fromIntegral p))
      root = (** (recip (fromIntegral p)))

-- | Dot (standard inner) product.
--
--   Examples:
--
--   >>> let v1 = make3d (1,2,3)
--   >>> let v2 = make3d (4,5,6)
--   >>> dot v1 v2
--   32
--
dot :: (Num a, Vector v a) => Vn (v a) -> Vn (v a) -> a
dot (Vn v1) (Vn v2) = V.sum $ V.zipWith (*) v1 v2


-- | The angle between @v1@ and @v2@ in Euclidean space.
--
--   Examples:
--
--   >>> let v1 = make2d (1.0, 0.0)
--   >>> let v2 = make2d (0.0, 1.0)
--   >>> angle v1 v2 == pi/2.0
--   True
--
angle :: (RealFloat a, Vector v a) => Vn (v a) -> Vn (v a) -> a
angle v1 v2 =
  acos theta
  where
    theta = (v1 `dot` v2) / norms
    norms = (norm_p (2 :: Integer) v1) * (norm_p (2 :: Integer) v2)


-- | Convenience function for creating 2d vectors.
--
--   Examples:
--
--   >>> let v1 = make2d (1,2)
--   >>> v1
--   (1,2)
--
make2d :: forall a. (a,a) -> Vn (Vec2 a)
make2d (x,y) =
  Vn v1
  where
    v1 = vec $ con |> x |> y :: Vec2 a


-- | Convenience function for creating 3d vectors.
--
--   Examples:
--
--   >>> let v1 = make3d (1,2,3)
--   >>> v1
--   (1,2,3)
--
make3d :: forall a. (a,a,a) -> Vn (Vec3 a)
make3d (x,y,z) =
  Vn v1
  where
    v1 = vec $ con |> x |> y |> z :: Vec3 a