Plucker.hs

{-# LANGUAGE CPP #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE GADTs #-}
#if __GLASGOW_HASKELL__ >= 702
{-# LANGUAGE Trustworthy #-}
{-# LANGUAGE DeriveGeneric #-}
#endif
#if __GLASGOW_HASKELL__ >= 707
{-# LANGUAGE DataKinds #-}
#endif

#if __GLASGOW_HASKELL__ >= 800
{-# LANGUAGE DeriveLift #-}
#endif

#ifndef MIN_VERSION_vector
#define MIN_VERSION_vector(x,y,z) 1
#endif

#ifndef MIN_VERSION_transformers
#define MIN_VERSION_transformers(x,y,z) 1
#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
--
-- Plücker coordinates for lines in 3d homogeneous space.
----------------------------------------------------------------------------
module Linear.Plucker
  ( Plucker(..)
  , squaredError
  , isotropic
  , (><)
  , plucker
  , plucker3D
  -- * Operations on lines
  , parallel
  , intersects
  , LinePass(..)
  , passes
  , quadranceToOrigin
  , closestToOrigin
  , isLine
  , coincides
  , coincides'
  -- * Basis elements
  ,      p01, p02, p03
  , p10,      p12, p13
  , p20, p21,      p23
  , p30, p31, p32

  , e01, e02, e03, e12, e31, e23
  ) where

import Control.Applicative
import Control.DeepSeq (NFData(rnf))
import Control.Monad (liftM)
import Control.Monad.Fix
import Control.Monad.Zip
import Control.Lens as Lens hiding (index, (<.>))
import Data.Binary as Binary
import Data.Bytes.Serial
import Data.Distributive
import Data.Foldable as Foldable
import qualified Data.Foldable.WithIndex as WithIndex
import Data.Functor.Bind
import Data.Functor.Classes
import Data.Functor.Rep
import qualified Data.Functor.WithIndex as WithIndex
import Data.Hashable
import Data.Semigroup
import Data.Semigroup.Foldable
import Data.Serialize as Cereal
import qualified Data.Traversable.WithIndex as WithIndex
#if __GLASGOW_HASKELL__ >= 707
import qualified Data.Vector as V
#endif
import qualified Data.Vector.Generic.Mutable as M
import qualified Data.Vector.Generic as G
import qualified Data.Vector.Unboxed.Base as U
import Foreign.Ptr (castPtr)
import Foreign.Storable (Storable(..))
import GHC.Arr (Ix(..))
#if __GLASGOW_HASKELL__ >= 702
import GHC.Generics (Generic)
#endif
#if __GLASGOW_HASKELL__ >= 706
import GHC.Generics (Generic1)
#endif
#if __GLASGOW_HASKELL__ >= 800
import Language.Haskell.TH.Syntax (Lift)
#endif
import Linear.Epsilon
import Linear.Metric
#if __GLASGOW_HASKELL__ >= 707
import Linear.V
#endif
import Linear.V2
import Linear.V3
import Linear.V4
import Linear.Vector
import System.Random

-- | Plücker coordinates for lines in a 3-dimensional space.
data Plucker a = Plucker !a !a !a !a !a !a deriving (Eq,Ord,Show,Read
#if __GLASGOW_HASKELL__ >= 702
                                                    ,Generic
#endif
#if __GLASGOW_HASKELL__ >= 706
                                                    ,Generic1
#endif
#if __GLASGOW_HASKELL__ >= 800
                                                    ,Lift
#endif
                                                    )

#if __GLASGOW_HASKELL__ >= 707
instance Finite Plucker where
  type Size Plucker = 6
  toV (Plucker a b c d e f) = V (V.fromListN 6 [a,b,c,d,e,f])
  fromV (V v) = Plucker (v V.! 0) (v V.! 1) (v V.! 2) (v V.! 3) (v V.! 4) (v V.! 5)
#endif

instance Random a => Random (Plucker a) where
  random g = case random g of
    (a, g1) -> case random g1 of
      (b, g2) -> case random g2 of
        (c, g3) -> case random g3 of
          (d, g4) -> case random g4 of
            (e, g5) -> case random g5 of
              (f, g6) -> (Plucker a b c d e f, g6)
  randomR (Plucker a b c d e f, Plucker a' b' c' d' e' f') g = case randomR (a,a') g of
    (a'', g1) -> case randomR (b,b') g1 of
      (b'', g2) -> case randomR (c,c') g2 of
        (c'', g3) -> case randomR (d,d') g3 of
          (d'', g4) -> case randomR (e,e') g4 of
            (e'', g5) -> case randomR (f,f') g5 of
              (f'', g6) -> (Plucker a'' b'' c'' d'' e'' f'', g6)

instance Functor Plucker where
  fmap g (Plucker a b c d e f) = Plucker (g a) (g b) (g c) (g d) (g e) (g f)
  {-# INLINE fmap #-}

instance Apply Plucker where
  Plucker a b c d e f <.> Plucker g h i j k l =
    Plucker (a g) (b h) (c i) (d j) (e k) (f l)
  {-# INLINE (<.>) #-}

instance Applicative Plucker where
  pure a = Plucker a a a a a a
  {-# INLINE pure #-}
  Plucker a b c d e f <*> Plucker g h i j k l =
    Plucker (a g) (b h) (c i) (d j) (e k) (f l)
  {-# INLINE (<*>) #-}

instance Additive Plucker where
  zero = pure 0
  {-# INLINE zero #-}
  liftU2 = liftA2
  {-# INLINE liftU2 #-}
  liftI2 = liftA2
  {-# INLINE liftI2 #-}

instance Bind Plucker where
  Plucker a b c d e f >>- g = Plucker a' b' c' d' e' f' where
    Plucker a' _ _ _ _ _ = g a
    Plucker _ b' _ _ _ _ = g b
    Plucker _ _ c' _ _ _ = g c
    Plucker _ _ _ d' _ _ = g d
    Plucker _ _ _ _ e' _ = g e
    Plucker _ _ _ _ _ f' = g f
  {-# INLINE (>>-) #-}

instance Monad Plucker where
  return a = Plucker a a a a a a
  {-# INLINE return #-}
  Plucker a b c d e f >>= g = Plucker a' b' c' d' e' f' where
    Plucker a' _ _ _ _ _ = g a
    Plucker _ b' _ _ _ _ = g b
    Plucker _ _ c' _ _ _ = g c
    Plucker _ _ _ d' _ _ = g d
    Plucker _ _ _ _ e' _ = g e
    Plucker _ _ _ _ _ f' = g f
  {-# INLINE (>>=) #-}

instance Distributive Plucker where
  distribute f = Plucker (fmap (\(Plucker x _ _ _ _ _) -> x) f)
                         (fmap (\(Plucker _ x _ _ _ _) -> x) f)
                         (fmap (\(Plucker _ _ x _ _ _) -> x) f)
                         (fmap (\(Plucker _ _ _ x _ _) -> x) f)
                         (fmap (\(Plucker _ _ _ _ x _) -> x) f)
                         (fmap (\(Plucker _ _ _ _ _ x) -> x) f)
  {-# INLINE distribute #-}

instance Representable Plucker where
  type Rep Plucker = E Plucker
  tabulate f = Plucker (f e01) (f e02) (f e03) (f e23) (f e31) (f e12)
  {-# INLINE tabulate #-}
  index xs (E l) = view l xs
  {-# INLINE index #-}

instance Foldable Plucker where
  foldMap g (Plucker a b c d e f) =
    g a `mappend` g b `mappend` g c `mappend` g d `mappend` g e `mappend` g f
  {-# INLINE foldMap #-}
#if __GLASGOW_HASKELL__ >= 710
  null _ = False
  length _ =  6
#endif

instance Traversable Plucker where
  traverse g (Plucker a b c d e f) =
    Plucker <$> g a <*> g b <*> g c <*> g d <*> g e <*> g f
  {-# INLINE traverse #-}

instance Foldable1 Plucker where
  foldMap1 g (Plucker a b c d e f) =
    g a <> g b <> g c <> g d <> g e <> g f
  {-# INLINE foldMap1 #-}

instance Traversable1 Plucker where
  traverse1 g (Plucker a b c d e f) =
    Plucker <$> g a <.> g b <.> g c <.> g d <.> g e <.> g f
  {-# INLINE traverse1 #-}

instance Ix a => Ix (Plucker a) where
  range (Plucker l1 l2 l3 l4 l5 l6,Plucker u1 u2 u3 u4 u5 u6) =
    [Plucker i1 i2 i3 i4 i5 i6 | i1 <- range (l1,u1)
                     , i2 <- range (l2,u2)
                     , i3 <- range (l3,u3)
                     , i4 <- range (l4,u4)
                     , i5 <- range (l5,u5)
                     , i6 <- range (l6,u6)
                     ]
  {-# INLINE range #-}

  unsafeIndex (Plucker l1 l2 l3 l4 l5 l6,Plucker u1 u2 u3 u4 u5 u6) (Plucker i1 i2 i3 i4 i5 i6) =
    unsafeIndex (l6,u6) i6 + unsafeRangeSize (l6,u6) * (
    unsafeIndex (l5,u5) i5 + unsafeRangeSize (l5,u5) * (
    unsafeIndex (l4,u4) i4 + unsafeRangeSize (l4,u4) * (
    unsafeIndex (l3,u3) i3 + unsafeRangeSize (l3,u3) * (
    unsafeIndex (l2,u2) i2 + unsafeRangeSize (l2,u2) *
    unsafeIndex (l1,u1) i1))))
  {-# INLINE unsafeIndex #-}

  inRange (Plucker l1 l2 l3 l4 l5 l6,Plucker u1 u2 u3 u4 u5 u6) (Plucker i1 i2 i3 i4 i5 i6) =
    inRange (l1,u1) i1 && inRange (l2,u2) i2 &&
    inRange (l3,u3) i3 && inRange (l4,u4) i4 &&
    inRange (l5,u5) i5 && inRange (l6,u6) i6
  {-# INLINE inRange #-}

instance Num a => Num (Plucker a) where
  (+) = liftA2 (+)
  {-# INLINE (+) #-}
  (-) = liftA2 (-)
  {-# INLINE (-) #-}
  (*) = liftA2 (*)
  {-# INLINE (*) #-}
  negate = fmap negate
  {-# INLINE negate #-}
  abs = fmap abs
  {-# INLINE abs #-}
  signum = fmap signum
  {-# INLINE signum #-}
  fromInteger = pure . fromInteger
  {-# INLINE fromInteger #-}

instance Fractional a => Fractional (Plucker a) where
  recip = fmap recip
  {-# INLINE recip #-}
  (/) = liftA2 (/)
  {-# INLINE (/) #-}
  fromRational = pure . fromRational
  {-# INLINE fromRational #-}

instance Floating a => Floating (Plucker a) where
    pi = pure pi
    {-# INLINE pi #-}
    exp = fmap exp
    {-# INLINE exp #-}
    sqrt = fmap sqrt
    {-# INLINE sqrt #-}
    log = fmap log
    {-# INLINE log #-}
    (**) = liftA2 (**)
    {-# INLINE (**) #-}
    logBase = liftA2 logBase
    {-# INLINE logBase #-}
    sin = fmap sin
    {-# INLINE sin #-}
    tan = fmap tan
    {-# INLINE tan #-}
    cos = fmap cos
    {-# INLINE cos #-}
    asin = fmap asin
    {-# INLINE asin #-}
    atan = fmap atan
    {-# INLINE atan #-}
    acos = fmap acos
    {-# INLINE acos #-}
    sinh = fmap sinh
    {-# INLINE sinh #-}
    tanh = fmap tanh
    {-# INLINE tanh #-}
    cosh = fmap cosh
    {-# INLINE cosh #-}
    asinh = fmap asinh
    {-# INLINE asinh #-}
    atanh = fmap atanh
    {-# INLINE atanh #-}
    acosh = fmap acosh
    {-# INLINE acosh #-}

instance Hashable a => Hashable (Plucker a) where
  hashWithSalt s (Plucker a b c d e f) = s `hashWithSalt` a `hashWithSalt` b `hashWithSalt` c `hashWithSalt` d `hashWithSalt` e `hashWithSalt` f
  {-# INLINE hashWithSalt #-}

instance Storable a => Storable (Plucker a) where
  sizeOf _ = 6 * sizeOf (undefined::a)
  {-# INLINE sizeOf #-}
  alignment _ = alignment (undefined::a)
  {-# INLINE alignment #-}
  poke ptr (Plucker a b c d e f) = do
    poke ptr' a
    pokeElemOff ptr' 1 b
    pokeElemOff ptr' 2 c
    pokeElemOff ptr' 3 d
    pokeElemOff ptr' 4 e
    pokeElemOff ptr' 5 f
    where ptr' = castPtr ptr
  {-# INLINE poke #-}
  peek ptr = Plucker <$> peek ptr'
                     <*> peekElemOff ptr' 1
                     <*> peekElemOff ptr' 2
                     <*> peekElemOff ptr' 3
                     <*> peekElemOff ptr' 4
                     <*> peekElemOff ptr' 5
    where ptr' = castPtr ptr
  {-# INLINE peek #-}

instance Metric Plucker where
  dot (Plucker a b c d e f) (Plucker g h i j k l) = a*g+b*h+c*i+d*j+e*k+f*l
  {-# INLINE dot #-}

instance Epsilon a => Epsilon (Plucker a) where
  nearZero = nearZero . quadrance
  {-# INLINE nearZero #-}

-- | Given a pair of points represented by homogeneous coordinates
-- generate Plücker coordinates for the line through them, directed
-- from the second towards the first.
plucker :: Num a => V4 a -> V4 a -> Plucker a
plucker (V4 a b c d)
        (V4 e f g h) =
  Plucker (a*f-b*e)
          (a*g-c*e)
          (b*g-c*f)
          (a*h-d*e)
          (b*h-d*f)
          (c*h-d*g)
{-# INLINE plucker #-}

-- | Given a pair of 3D points, generate Plücker coordinates for the
-- line through them, directed from the second towards the first.
plucker3D :: Num a => V3 a -> V3 a -> Plucker a
plucker3D p q = Plucker a b c d e f
  where V3 a b c = p - q
        V3 d e f = p `cross` q

-- | These elements form a basis for the Plücker space, or the Grassmanian manifold @Gr(2,V4)@.
--
-- @
-- 'p01' :: 'Lens'' ('Plucker' a) a
-- 'p02' :: 'Lens'' ('Plucker' a) a
-- 'p03' :: 'Lens'' ('Plucker' a) a
-- 'p23' :: 'Lens'' ('Plucker' a) a
-- 'p31' :: 'Lens'' ('Plucker' a) a
-- 'p12' :: 'Lens'' ('Plucker' a) a
-- @
p01, p02, p03, p23, p31, p12 :: Lens' (Plucker a) a
p01 g (Plucker a b c d e f) = (\a' -> Plucker a' b c d e f) <$> g a
p02 g (Plucker a b c d e f) = (\b' -> Plucker a b' c d e f) <$> g b
p03 g (Plucker a b c d e f) = (\c' -> Plucker a b c' d e f) <$> g c
p23 g (Plucker a b c d e f) = (\d' -> Plucker a b c d' e f) <$> g d
p31 g (Plucker a b c d e f) = (\e' -> Plucker a b c d e' f) <$> g e
p12 g (Plucker a b c d e f) = Plucker a b c d e <$> g f
{-# INLINE p01 #-}
{-# INLINE p02 #-}
{-# INLINE p03 #-}
{-# INLINE p23 #-}
{-# INLINE p31 #-}
{-# INLINE p12 #-}

-- | These elements form an alternate basis for the Plücker space, or the Grassmanian manifold @Gr(2,V4)@.
--
-- @
-- 'p10' :: 'Num' a => 'Lens'' ('Plucker' a) a
-- 'p20' :: 'Num' a => 'Lens'' ('Plucker' a) a
-- 'p30' :: 'Num' a => 'Lens'' ('Plucker' a) a
-- 'p32' :: 'Num' a => 'Lens'' ('Plucker' a) a
-- 'p13' :: 'Num' a => 'Lens'' ('Plucker' a) a
-- 'p21' :: 'Num' a => 'Lens'' ('Plucker' a) a
-- @
p10, p20, p30, p32, p13, p21 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
p10 = anti p01
p20 = anti p02
p30 = anti p03
p32 = anti p23
p13 = anti p31
p21 = anti p21
{-# INLINE p10 #-}
{-# INLINE p20 #-}
{-# INLINE p30 #-}
{-# INLINE p32 #-}
{-# INLINE p13 #-}
{-# INLINE p21 #-}

anti :: (Functor f, Num a) => ((a -> f a) -> r) -> (a -> f a) -> r
anti k f = k (fmap negate . f . negate)

e01, e02, e03, e23, e31, e12 :: E Plucker
e01 = E p01
e02 = E p02
e03 = E p03
e23 = E p23
e31 = E p31
e12 = E p12

instance WithIndex.FunctorWithIndex (E Plucker) Plucker where
  imap f (Plucker a b c d e g) = Plucker (f e01 a) (f e02 b) (f e03 c) (f e23 d) (f e31 e) (f e12 g)
  {-# INLINE imap #-}

instance WithIndex.FoldableWithIndex (E Plucker) Plucker where
  ifoldMap f (Plucker a b c d e g) = f e01 a `mappend` f e02 b `mappend` f e03 c
                           `mappend` f e23 d `mappend` f e31 e `mappend` f e12 g
  {-# INLINE ifoldMap #-}

instance WithIndex.TraversableWithIndex (E Plucker) Plucker where
  itraverse f (Plucker a b c d e g) = Plucker <$> f e01 a <*> f e02 b <*> f e03 c
                                              <*> f e23 d <*> f e31 e <*> f e12 g
  {-# INLINE itraverse #-}

#if !MIN_VERSION_lens(5,0,0)
instance Lens.FunctorWithIndex     (E Plucker) Plucker where imap      = WithIndex.imap
instance Lens.FoldableWithIndex    (E Plucker) Plucker where ifoldMap  = WithIndex.ifoldMap
instance Lens.TraversableWithIndex (E Plucker) Plucker where itraverse = WithIndex.itraverse
#endif

type instance Index (Plucker a) = E Plucker
type instance IxValue (Plucker a) = a

instance Ixed (Plucker a) where
  ix i = el i
  {-# INLINE ix #-}

instance Each (Plucker a) (Plucker b) a b where
  each = traverse
  {-# INLINE each #-}


-- | Valid Plücker coordinates @p@ will have @'squaredError' p '==' 0@
--
-- That said, floating point makes a mockery of this claim, so you may want to use 'nearZero'.
squaredError :: Num a => Plucker a -> a
squaredError v = v >< v
{-# INLINE squaredError #-}

-- | This isn't th actual metric because this bilinear form gives rise to an isotropic quadratic space
infixl 5 ><
(><) :: Num a => Plucker a -> Plucker a -> a
Plucker a b c d e f >< Plucker g h i j k l = a*l-b*k+c*j+d*i-e*h+f*g
{-# INLINE (><) #-}

-- | Checks if the line is near-isotropic (isotropic vectors in this
-- quadratic space represent lines in real 3d space).
isotropic :: Epsilon a => Plucker a -> Bool
isotropic a = nearZero (a >< a)
{-# INLINE isotropic #-}

-- | Checks if two lines intersect (or nearly intersect).
intersects :: (Epsilon a, Ord a) => Plucker a -> Plucker a -> Bool
intersects a b = not (a `parallel` b) && passes a b == Coplanar
-- intersects :: Epsilon a => Plucker a -> Plucker a -> Bool
-- intersects a b = nearZero (a >< b)
{-# INLINE intersects #-}

-- | Describe how two lines pass each other.
data LinePass = Coplanar
              -- ^ The lines are coplanar (parallel or intersecting).
              | Clockwise
              -- ^ The lines pass each other clockwise (right-handed
              -- screw)
              | Counterclockwise
              -- ^ The lines pass each other counterclockwise
              -- (left-handed screw).
                deriving (Eq, Show
#if defined(__GLASGOW_HASKELL__) && __GLASGOW_HASKELL__ >= 702
                         ,Generic
#endif
                         )

-- | Check how two lines pass each other. @passes l1 l2@ describes
-- @l2@ when looking down @l1@.
passes :: (Epsilon a, Ord a) => Plucker a -> Plucker a -> LinePass
passes a b
  | nearZero s = Coplanar
  | s > 0 = Counterclockwise
  | otherwise = Clockwise
  where s = (u1 `dot` v2) + (u2 `dot` v1)
        V2 u1 v1 = toUV a
        V2 u2 v2 = toUV b
{-# INLINE passes #-}

-- | Checks if two lines are parallel.
parallel :: Epsilon a => Plucker a -> Plucker a -> Bool
parallel a b = nearZero $ u1 `cross` u2
  where V2 u1 _ = toUV a
        V2 u2 _ = toUV b
{-# INLINE parallel #-}

-- | Represent a Plücker coordinate as a pair of 3-tuples, typically
-- denoted U and V.
toUV :: Plucker a -> V2 (V3 a)
toUV (Plucker a b c d e f) = V2 (V3 a b c) (V3 d e f)

-- | Checks if two lines coincide in space. In other words, undirected equality.
coincides :: (Epsilon a, Fractional a) => Plucker a -> Plucker a -> Bool
coincides p1 p2 = Foldable.all nearZero $ (s *^ p2) - p1
  where s = maybe 1 getFirst . getOptionCompat . fold $ saveDiv <$> p1 <*> p2
        saveDiv x y | nearZero y = optionCompat Nothing
                    | otherwise  = optionCompat . Just $ First (x / y)
{-# INLINABLE coincides #-}

-- | Checks if two lines coincide in space, and have the same
-- orientation.
coincides' :: (Epsilon a, Fractional a, Ord a) => Plucker a -> Plucker a -> Bool
coincides' p1 p2 = Foldable.all nearZero ((s *^ p2) - p1) && s > 0
  where s = maybe 1 getFirst . getOptionCompat . fold $ saveDiv <$> p1 <*> p2
        saveDiv x y | nearZero y = optionCompat Nothing
                    | otherwise  = optionCompat . Just $ First (x / y)
{-# INLINABLE coincides' #-}

-- The coincides and coincides' functions above require the use of a Maybe type
-- with the following Monoid instance:
--
--   instance Semigroup a => Monoid (Maybe a) where ...
--
-- Unfortunately, Maybe has only had such an instance since base-4.11. Prior
-- to that, its Monoid instance had an instance context of Monoid a, which is
-- too strong. To compensate, we use CPP to define an OptionCompat type
-- synonym, which is an alias for Maybe on recent versions of base and an alias
-- for Data.Semigroup.Option on older versions of base. We don't want to use
-- Option on recent versions of base, as it is deprecated.
#if MIN_VERSION_base(4,11,0)
type OptionCompat = Maybe

optionCompat :: Maybe a -> OptionCompat a
optionCompat = id

getOptionCompat :: OptionCompat a -> Maybe a
getOptionCompat = id
#else
type OptionCompat = Option

optionCompat :: Maybe a -> OptionCompat a
optionCompat = Option

getOptionCompat :: OptionCompat a -> Maybe a
getOptionCompat = getOption
#endif

-- | The minimum squared distance of a line from the origin.
quadranceToOrigin :: Fractional a => Plucker a -> a
quadranceToOrigin p = (v `dot` v) / (u `dot` u)
  where V2 u v = toUV p
{-# INLINE quadranceToOrigin #-}

-- | The point where a line is closest to the origin.
closestToOrigin :: Fractional a => Plucker a -> V3 a
closestToOrigin p = normalizePoint $ V4 x y z (u `dot` u)
  where V2 u v = toUV p
        V3 x y z = v `cross` u
{-# INLINE closestToOrigin #-}

-- | Not all 6-dimensional points correspond to a line in 3D. This
-- predicate tests that a Plücker coordinate lies on the Grassmann
-- manifold, and does indeed represent a 3D line.
isLine :: Epsilon a => Plucker a -> Bool
isLine p = nearZero $ u `dot` v
  where V2 u v = toUV p
{-# INLINE isLine #-}

-- TODO: drag some more stuff out of my thesis

data instance U.Vector    (Plucker a) =  V_Plucker !Int (U.Vector    a)
data instance U.MVector s (Plucker a) = MV_Plucker !Int (U.MVector s a)
instance U.Unbox a => U.Unbox (Plucker a)

instance U.Unbox a => M.MVector U.MVector (Plucker a) where
  basicLength (MV_Plucker n _) = n
  basicUnsafeSlice m n (MV_Plucker _ v) = MV_Plucker n (M.basicUnsafeSlice (6*m) (6*n) v)
  basicOverlaps (MV_Plucker _ v) (MV_Plucker _ u) = M.basicOverlaps v u
  basicUnsafeNew n = liftM (MV_Plucker n) (M.basicUnsafeNew (6*n))
  basicUnsafeRead (MV_Plucker _ a) i =
    do let o = 6*i
       x <- M.basicUnsafeRead a o
       y <- M.basicUnsafeRead a (o+1)
       z <- M.basicUnsafeRead a (o+2)
       w <- M.basicUnsafeRead a (o+3)
       v <- M.basicUnsafeRead a (o+4)
       u <- M.basicUnsafeRead a (o+5)
       return (Plucker x y z w v u)
  basicUnsafeWrite (MV_Plucker _ a) i (Plucker x y z w v u) =
    do let o = 6*i
       M.basicUnsafeWrite a o     x
       M.basicUnsafeWrite a (o+1) y
       M.basicUnsafeWrite a (o+2) z
       M.basicUnsafeWrite a (o+3) w
       M.basicUnsafeWrite a (o+4) v
       M.basicUnsafeWrite a (o+5) u
#if MIN_VERSION_vector(0,11,0)
  basicInitialize (MV_Plucker _ v) = M.basicInitialize v
#endif

instance U.Unbox a => G.Vector U.Vector (Plucker a) where
  basicUnsafeFreeze (MV_Plucker n v) = liftM ( V_Plucker n) (G.basicUnsafeFreeze v)
  basicUnsafeThaw   ( V_Plucker n v) = liftM (MV_Plucker n) (G.basicUnsafeThaw   v)
  basicLength       ( V_Plucker n _) = n
  basicUnsafeSlice m n (V_Plucker _ v) = V_Plucker n (G.basicUnsafeSlice (6*m) (6*n) v)
  basicUnsafeIndexM (V_Plucker _ a) i =
    do let o = 6*i
       x <- G.basicUnsafeIndexM a o
       y <- G.basicUnsafeIndexM a (o+1)
       z <- G.basicUnsafeIndexM a (o+2)
       w <- G.basicUnsafeIndexM a (o+3)
       v <- G.basicUnsafeIndexM a (o+4)
       u <- G.basicUnsafeIndexM a (o+5)
       return (Plucker x y z w v u)

instance MonadZip Plucker where
  mzipWith = liftA2

instance MonadFix Plucker where
  mfix f = Plucker (let Plucker a _ _ _ _ _ = f a in a)
                   (let Plucker _ a _ _ _ _ = f a in a)
                   (let Plucker _ _ a _ _ _ = f a in a)
                   (let Plucker _ _ _ a _ _ = f a in a)
                   (let Plucker _ _ _ _ a _ = f a in a)
                   (let Plucker _ _ _ _ _ a = f a in a)

instance NFData a => NFData (Plucker a) where
  rnf (Plucker a b c d e f) = rnf a `seq` rnf b `seq` rnf c
                        `seq` rnf d `seq` rnf e `seq` rnf f

instance Serial1 Plucker where
  serializeWith = traverse_
  deserializeWith k = Plucker <$> k <*> k <*> k <*> k <*> k <*> k

instance Serial a => Serial (Plucker a) where
  serialize = serializeWith serialize
  deserialize = deserializeWith deserialize

instance Binary a => Binary (Plucker a) where
  put = serializeWith Binary.put
  get = deserializeWith Binary.get

instance Serialize a => Serialize (Plucker a) where
  put = serializeWith Cereal.put
  get = deserializeWith Cereal.get


#if (MIN_VERSION_transformers(0,5,0)) || !(MIN_VERSION_transformers(0,4,0))
instance Eq1 Plucker where
  liftEq k (Plucker a1 b1 c1 d1 e1 f1)
           (Plucker a2 b2 c2 d2 e2 f2)
         = k a1 a2 && k b1 b2 && k c1 c2 && k d1 d2 && k e1 e2 && k f1 f2
instance Ord1 Plucker where
  liftCompare k (Plucker a1 b1 c1 d1 e1 f1)
                (Plucker a2 b2 c2 d2 e2 f2)
            = k a1 a2 `mappend` k b1 b2 `mappend` k c1 c2 `mappend` k d1 d2 `mappend` k e1 e2 `mappend` k f1 f2
instance Read1 Plucker where
  liftReadsPrec k _ z = readParen (z > 10) $ \r ->
     [ (Plucker a b c d e f, r7)
     | ("Plucker",r1) <- lex r
     , (a,r2) <- k 11 r1
     , (b,r3) <- k 11 r2
     , (c,r4) <- k 11 r3
     , (d,r5) <- k 11 r4
     , (e,r6) <- k 11 r5
     , (f,r7) <- k 11 r6
     ]
instance Show1 Plucker where
  liftShowsPrec k _ z (Plucker a b c d e f) = showParen (z > 10) $
     showString "Plucker " . k 11 a . showChar ' ' . k 11 b . showChar ' ' . k 11 c . showChar ' ' . k 11 d . showChar ' ' . k 11 e . showChar ' ' . k 11 f
#else
instance Eq1 Plucker where eq1 = (==)
instance Ord1 Plucker where compare1 = compare
instance Show1 Plucker where showsPrec1 = showsPrec
instance Read1 Plucker where readsPrec1 = readsPrec
#endif

instance Field1 (Plucker a) (Plucker a) a a where
  _1 f (Plucker x y z u v w) = f x <&> \x' -> Plucker x' y z u v w

instance Field2 (Plucker a) (Plucker a) a a where
  _2 f (Plucker x y z u v w) = f y <&> \y' -> Plucker x y' z u v w

instance Field3 (Plucker a) (Plucker a) a a where
  _3 f (Plucker x y z u v w) = f z <&> \z' -> Plucker x y z' u v w

instance Field4 (Plucker a) (Plucker a) a a where
  _4 f (Plucker x y z u v w) = f u <&> \u' -> Plucker x y z u' v w

instance Field5 (Plucker a) (Plucker a) a a where
  _5 f (Plucker x y z u v w) = f v <&> \v' -> Plucker x y z u v' w

instance Field6 (Plucker a) (Plucker a) a a where
  _6 f (Plucker x y z u v w) = f w <&> \w' -> Plucker x y z u v w'

instance Semigroup a => Semigroup (Plucker a) where
 (<>) = liftA2 (<>)

instance Monoid a => Monoid (Plucker a) where
  mempty = pure mempty
#if !(MIN_VERSION_base(4,11,0))
  mappend = liftA2 mappend
#endif