    {-# LANGUAGE FlexibleContexts     #-}
    {-# LANGUAGE TypeFamilies         #-}
    {-# LANGUAGE TypeOperators        #-}
    {-# LANGUAGE UndecidableInstances #-}
    -----------------------------------------------------------------------------
    -- |
    -- Module      :  Diagrams.Coordinates
    -- Copyright   :  (c) 2012 diagrams-lib team (see LICENSE)
    -- License     :  BSD-style (see LICENSE)
    -- Maintainer  :  diagrams-discuss@googlegroups.com
    --
    -- Nice syntax for constructing and pattern-matching on literal
    -- points and vectors.
    --
    -----------------------------------------------------------------------------

    module Diagrams.Coordinates
        ( (:&)(..), Coordinates(..)
        )
        where

    import           Diagrams.Points

    import           Linear          (V2 (..), V3 (..), V4 (..))

    -- | Types which are instances of the @Coordinates@ class can be
    --   constructed using '^&' (for example, a three-dimensional vector
    --   could be constructed by @1 ^& 6 ^& 3@), and deconstructed using
    --   'coords'.  A common pattern is to use 'coords' in conjunction
    --   with the @ViewPatterns@ extension, like so:
    --
    -- @
    -- foo :: Vector3 -> ...
    -- foo (coords -> x :& y :& z) = ...
    -- @
    class Coordinates c where

      -- | The type of the final coordinate.
      type FinalCoord c    :: *

      -- | The type of everything other than the final coordinate.
      type PrevDim c       :: *

      -- | Decomposition of @c@ into applications of ':&'.
      type Decomposition c :: *
        -- Decomposition c = Decomposition (PrevDim c) :& FinalCoord c  (essentially)

      -- | Construct a value of type @c@ by providing something of one
      --   less dimension (which is perhaps itself recursively constructed
      --   using @(^&)@) and a final coordinate.  For example,
      --
      -- @
      -- 2 ^& 3 :: P2
      -- 3 ^& 5 ^& 6 :: V3
      -- @
      --
      --   Note that @^&@ is left-associative.
      (^&)    :: PrevDim c -> FinalCoord c -> c

      -- | Prefix synonym for @^&@. pr stands for pair of @PrevDim@, @FinalCoord@
      pr      :: PrevDim c -> FinalCoord c -> c
      pr = (^&)

      -- | Decompose a value of type @c@ into its constituent coordinates,
      --   stored in a nested @(:&)@ structure.
      coords :: c -> Decomposition c

    infixl 7 ^&

    -- | A pair of values, with a convenient infix (left-associative)
    --   data constructor.
    data a :& b = a :& b
      deriving (Eq, Ord, Show)

    infixl 7 :&

    -- Instance for :& (the buck stops here)
    instance Coordinates (a :& b) where
      type FinalCoord (a :& b) = b
      type PrevDim (a :& b) = a
      type Decomposition (a :& b) = a :& b
      x ^& y                    = x :& y
      coords (x :& y)           = x :& y

    -- Some standard instances for plain old tuples

    instance Coordinates (a,b) where
      type FinalCoord (a,b)    = b
      type PrevDim (a,b)       = a
      type Decomposition (a,b) = a :& b

      x ^& y                   = (x,y)
      coords (x,y)             = x :& y

    instance Coordinates (a,b,c) where
      type FinalCoord (a,b,c)    = c
      type PrevDim (a,b,c)       = (a,b)
      type Decomposition (a,b,c) = Decomposition (a,b) :& c

      (x,y) ^& z                  = (x,y,z)
      coords (x,y,z)             = coords (x,y) :& z

    instance Coordinates (a,b,c,d) where
      type FinalCoord (a,b,c,d)    = d
      type PrevDim (a,b,c,d)       = (a,b,c)
      type Decomposition (a,b,c,d) = Decomposition (a,b,c) :& d

      (w,x,y)  ^& z                = (w,x,y,z)
      coords (w,x,y,z)             = coords (w,x,y) :& z

    instance Coordinates (v n) => Coordinates (Point v n) where
      type FinalCoord (Point v n)    = FinalCoord (v n)
      type PrevDim (Point v n)       = PrevDim (v n)
      type Decomposition (Point v n) = Decomposition (v n)

      x ^& y       = P (x ^& y)
      coords (P v) = coords v

    -- instances for linear

    instance Coordinates (V2 n) where
      type FinalCoord (V2 n)    = n
      type PrevDim (V2 n)       = n
      type Decomposition (V2 n) = n :& n

      x ^& y          = V2 x y
      coords (V2 x y) = x :& y

    instance Coordinates (V3 n) where
      type FinalCoord (V3 n)    = n
      type PrevDim (V3 n)       = V2 n
      type Decomposition (V3 n) = n :& n :& n

      V2 x y ^& z       = V3 x y z
      coords (V3 x y z) = x :& y :& z

    instance Coordinates (V4 n) where
      type FinalCoord (V4 n)    = n
      type PrevDim (V4 n)       = V3 n
      type Decomposition (V4 n) = n :& n :& n :& n

      V3 x y z ^& w       = V4 x y z w
      coords (V4 x y z w) = x :& y :& z :& w