```{-# LANGUAGE DefaultSignatures #-}
-----------------------------------------------------------------------------
-- |
-- Module      :  Linear.Metric
-- Copyright   :  (C) 2012-2013 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
) where

import Data.Foldable as Foldable
import Data.Functor.Apply
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
default dot :: (Foldable f, Num a) => f a -> f a -> a
dot x y = Foldable.sum \$ liftF2 (*) x y

-- | 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

-- | 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
```