```{-# LANGUAGE CPP #-}
{-# LANGUAGE DefaultSignatures #-}
{-# LANGUAGE Trustworthy #-}
#endif
-----------------------------------------------------------------------------
-- |
-- Copyright   :  (C) 2012-2015 Edward Kmett
-- License     :  BSD-style (see the file LICENSE)
--
-- Maintainer  :  Edward Kmett <ekmett@gmail.com>
-- Stability   :  experimental
-- Portability :  non-portable
--
-- Free metric spaces
----------------------------------------------------------------------------
module Linear.Metric
( Metric(..), normalize, project
) where

import Control.Applicative
import Data.Foldable as Foldable
import Data.Functor.Identity
import Data.Vector (Vector)
import Data.IntMap (IntMap)
import Data.Map (Map)
import Data.HashMap.Strict (HashMap)
import Data.Hashable (Hashable)
import Linear.Epsilon
import Linear.Vector

-- \$setup
-- >>> import Linear

-- | Free and sparse inner product/metric spaces.
class Additive f => Metric f where
-- | Compute the inner product of two vectors or (equivalently)
-- convert a vector @f a@ into a covector @f a -> a@.
--
-- >>> V2 1 2 `dot` V2 3 4
-- 11
dot :: Num a => f a -> f a -> a
#ifndef HLINT
default dot :: (Foldable f, Num a) => f a -> f a -> a
dot x y = Foldable.sum \$ liftI2 (*) x y
#endif

-- | Compute the squared norm. The name quadrance arises from
-- Norman J. Wildberger's rational trigonometry.
quadrance :: Num a => f a -> a
quadrance v = dot v v

-- | Compute the quadrance of the difference
qd :: Num a => f a -> f a -> a
qd f g = quadrance (f ^-^ g)

-- | Compute the distance between two vectors in a metric space
distance :: Floating a => f a -> f a -> a
distance f g = norm (f ^-^ g)

-- | Compute the norm of a vector in a metric space
norm :: Floating a => f a -> a
norm v = sqrt (quadrance v)

-- | Convert a non-zero vector to unit vector.
signorm :: Floating a => f a -> f a
signorm v = fmap (/m) v where
m = norm v

instance Metric Identity where
dot (Identity x) (Identity y) = x * y

instance Metric []

instance Metric Maybe

instance Metric ZipList where
-- ZipList is missing its Foldable instance
dot (ZipList x) (ZipList y) = dot x y

instance Metric IntMap

instance Ord k => Metric (Map k)

instance (Hashable k, Eq k) => Metric (HashMap k)

instance Metric Vector

-- | Normalize a 'Metric' functor to have unit 'norm'. This function
-- does not change the functor if its 'norm' is 0 or 1.
normalize :: (Floating a, Metric f, Epsilon a) => f a -> f a
normalize v = if nearZero l || nearZero (1-l) then v else fmap (/sqrt l) v
where l = quadrance v

-- | @project u v@ computes the projection of @v@ onto @u@.
project :: (Metric v, Fractional a) => v a -> v a -> v a
project u v = ((v `dot` u) / quadrance u) *^ u

```