```{-# LANGUAGE TypeFamilies  #-}
{-# LANGUAGE TypeOperators #-}
-----------------------------------------------------------------------------
-- |
-- 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(..)

-- * Lenses for particular axes
, HasX(..), HasY(..), HasZ(..), HasR(..)
)
where

import           Control.Lens          (Lens')

import           Diagrams.Core.Points

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

infixl 7 :&

-- | 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 :: R3
-- @
--
--   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 ^&

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

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

-- | The class of types with at least one coordinate, called _x.
class HasX t where
_x :: Lens' t Double

-- | The class of types with at least two coordinates, the second called _y.
class HasY t where
_y :: Lens' t Double

-- | The class of types with at least three coordinates, the third called _z.
class HasZ t where
_z :: Lens' t Double

-- | The class of types with a single length coordinate _r.  _r is
-- magnitude of a vector, or the distance from the origin of a point.
class HasR t where
_r :: Lens' t Double
```