```{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE TypeFamilies     #-}
{-# LANGUAGE ViewPatterns     #-}
-----------------------------------------------------------------------------
-- |
-- Module      :  Diagrams.TwoD.Vector
-- Copyright   :  (c) 2011 diagrams-lib team (see LICENSE)
-- License     :  BSD-style (see LICENSE)
-- Maintainer  :  diagrams-discuss@googlegroups.com
--
-- Two-dimensional vectors.
--
-----------------------------------------------------------------------------
module Diagrams.TwoD.Vector
( -- * Special 2D vectors
unitX, unitY, unit_X, unit_Y

-- * Converting between vectors and angles
, direction, angleBetween, fromDirection, e

-- * 2D vector utilities
, perp, leftTurn
) where

import           Control.Lens         ((^.))
import           Data.AdditiveGroup
import           Data.VectorSpace     ((<.>))
import           Diagrams.TwoD.Types
import           Diagrams.Coordinates

-- | The unit vector in the positive X direction.
unitX :: R2
unitX = 1 ^& 0

-- | The unit vector in the positive Y direction.
unitY :: R2
unitY = 0 ^& 1

-- | The unit vector in the negative X direction.
unit_X :: R2
unit_X = (-1) ^& 0

-- | The unit vector in the negative Y direction.
unit_Y :: R2
unit_Y = 0 ^& (-1)

-- | Compute the direction of a vector, measured counterclockwise from
--   the positive x-axis as a fraction of a full turn.  The zero
--   vector is arbitrarily assigned the direction 0.
direction :: R2 -> Angle
direction (coords -> x :& y) = atan2 y x @@ rad

-- | Compute the counterclockwise angle from the first vector to the second.
angleBetween :: R2 -> R2 -> Angle
angleBetween v1 v2
| d2 > d1   = d2 ^-^ d1
| otherwise = fullTurn ^+^ d2 ^-^ d1
where
d1 = direction v1
d2 = direction v2

-- | Convert an angle into a unit vector pointing in that direction.
fromDirection :: Angle -> R2
fromDirection a = cos a' ^& sin a'
where a' = a^.rad

-- | A convenient synonym for 'fromDirection'.
e :: Angle -> R2
e = fromDirection

-- | @perp v@ is perpendicular to and has the same magnitude as @v@.
--   In particular @perp v == rotateBy (1/4) v@.
perp :: R2 -> R2
perp (coords -> x :& y) = (-y) ^& x

-- | @leftTurn v1 v2@ tests whether the direction of @v2@ is a left
--   turn from @v1@ (that is, if the direction of @v2@ can be obtained
--   from that of @v1@ by adding an angle 0 <= theta <= tau/2).
leftTurn :: R2 -> R2 -> Bool
leftTurn v1 v2 = (v1 <.> perp v2) < 0
```