{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE BangPatterns               #-}
{-# LANGUAGE DeriveFunctor              #-}
{-# LANGUAGE EmptyDataDecls             #-}
{-# LANGUAGE FlexibleContexts           #-}
{-# LANGUAGE FlexibleInstances          #-}
{-# LANGUAGE GADTs                      #-}
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE MultiParamTypeClasses      #-}
{-# LANGUAGE StandaloneDeriving         #-}
{-# LANGUAGE TemplateHaskell            #-}
{-# LANGUAGE TypeFamilies               #-}
{-# LANGUAGE TypeOperators              #-}
{-# LANGUAGE UndecidableInstances       #-}

-----------------------------------------------------------------------------
-- |
-- Module      :  Diagrams.Segment
-- Copyright   :  (c) 2011-2013 diagrams-lib team (see LICENSE)
-- License     :  BSD-style (see LICENSE)
-- Maintainer  :  diagrams-discuss@googlegroups.com
--
-- A /segment/ is a translation-invariant, atomic path.  Currently,
-- there are two types: linear (/i.e./ just a straight line to the
-- endpoint) and cubic Bézier curves (/i.e./ a curve to an endpoint
-- with two control points).  This module contains tools for creating
-- and manipulating segments, as well as a definition of segments with
-- a fixed location (useful for backend implementors).
--
-- Generally speaking, casual users of diagrams should not need this
-- module; the higher-level functionality provided by
-- "Diagrams.Trail", "Diagrams.TrailLike", and "Diagrams.Path" should
-- usually suffice.  However, directly manipulating segments can
-- occasionally be useful.
--
-----------------------------------------------------------------------------

module Diagrams.Segment
       ( -- * Open/closed tags

         Open, Closed

         -- * Segment offsets

       , Offset(..) , segOffset

         -- * Constructing and modifying segments

       , Segment(..), straight, bezier3, bézier3, reverseSegment, mapSegmentVectors
       , openLinear, openCubic

         -- * Fixed (absolutely located) segments
       , FixedSegment(..)
       , mkFixedSeg, fromFixedSeg
       , fixedSegIso

         -- * Segment measures
         -- $segmeas

       , SegCount(..)
       , ArcLength(..)
       , getArcLengthCached, getArcLengthFun, getArcLengthBounded
       , TotalOffset(..)
       , OffsetEnvelope(..), oeOffset, oeEnvelope
       , SegMeasure

       ) where

import           Control.Lens              hiding (at, transform)
import           Data.FingerTree
import           Data.Monoid.MList
import           Data.Semigroup
import           Numeric.Interval.Kaucher  (Interval (..))
import qualified Numeric.Interval.Kaucher  as I

import           Linear.Affine
import           Linear.Metric
import           Linear.Vector             hiding (lerp)
import           Linear.Vector.Compat      (lerp)

import           Control.Applicative
import           Diagrams.Core             hiding (Measured)
import           Diagrams.Located
import           Diagrams.Parametric
import           Diagrams.Solve.Polynomial

import           Data.Serialize            (Serialize)
import qualified Data.Serialize            as Serialize

------------------------------------------------------------
--  Open/closed type tags  ---------------------------------
------------------------------------------------------------

-- Eventually we should use DataKinds for this, but not until we drop
-- support for GHC 7.4.

-- | Type tag for open segments.
data Open

-- | Type tag for closed segments.
data Closed

------------------------------------------------------------
--  Segment offsets  ---------------------------------------
------------------------------------------------------------

-- | The /offset/ of a segment is the vector from its starting point
--   to its end.  The offset for an /open/ segment is determined by
--   the context, /i.e./ its endpoint is not fixed.  The offset for a
--   /closed/ segment is stored explicitly, /i.e./ its endpoint is at
--   a fixed offset from its start.
data Offset c v n where
  OffsetOpen   :: Offset Open v n
  OffsetClosed :: v n -> Offset Closed v n

deriving instance Show (v n) => Show (Offset c v n)
deriving instance Eq   (v n) => Eq   (Offset c v n)
deriving instance Ord  (v n) => Ord  (Offset c v n)

instance Functor v => Functor (Offset c v) where
  fmap _ OffsetOpen       = OffsetOpen
  fmap f (OffsetClosed v) = OffsetClosed (fmap f v)

instance Each (Offset c v n) (Offset c v' n') (v n) (v' n') where
  each f (OffsetClosed v) = OffsetClosed <$> f v
  each _ OffsetOpen       = pure OffsetOpen
  {-# INLINE each #-}

-- | Reverses the direction of closed offsets.
instance (Additive v, Num n) => Reversing (Offset c v n) where
  reversing (OffsetClosed off) = OffsetClosed $ negated off
  reversing a@OffsetOpen       = a

type instance V (Offset c v n) = v
type instance N (Offset c v n) = n

instance Transformable (Offset c v n) where
  transform _ OffsetOpen       = OffsetOpen
  transform t (OffsetClosed v) = OffsetClosed (apply t v)

------------------------------------------------------------
--  Constructing segments  ---------------------------------
------------------------------------------------------------

-- | The atomic constituents of the concrete representation currently
--   used for trails are /segments/, currently limited to
--   single straight lines or cubic Bézier curves.  Segments are
--   /translationally invariant/, that is, they have no particular
--   \"location\" and are unaffected by translations.  They are,
--   however, affected by other transformations such as rotations and
--   scales.
data Segment c v n
    = Linear !(Offset c v n)
      -- ^ A linear segment with given offset.

    | Cubic !(v n) !(v n) !(Offset c v n)
      -- ^ A cubic Bézier segment specified by
      --   three offsets from the starting
      --   point to the first control point,
      --   second control point, and ending
      --   point, respectively.

  deriving (Functor, Eq, Ord)

instance Show (v n) => Show (Segment c v n) where
  showsPrec d seg = case seg of
    Linear (OffsetClosed v)       -> showParen (d > 10) $
      showString "straight " . showsPrec 11 v
    Cubic v1 v2 (OffsetClosed v3) -> showParen (d > 10) $
      showString "bézier3  " . showsPrec 11 v1 . showChar ' '
                             . showsPrec 11 v2 . showChar ' '
                             . showsPrec 11 v3
    Linear OffsetOpen             -> showString "openLinear"
    Cubic v1 v2 OffsetOpen        -> showParen (d > 10) $
      showString "openCubic " . showsPrec 11 v1 . showChar ' '
                              . showsPrec 11 v2


instance Each (Segment c v n) (Segment c v' n') (v n) (v' n') where
  each f (Linear offset)      = Linear <$> each f offset
  each f (Cubic v1 v2 offset) = Cubic  <$> f v1 <*> f v2 <*> each f offset
  {-# INLINE each #-}

-- | Reverse the direction of a segment.
instance (Additive v, Num n) => Reversing (Segment Closed v n) where
  reversing = reverseSegment

-- | Map over the vectors of each segment.
mapSegmentVectors :: (v n -> v' n') -> Segment c v n -> Segment c v' n'
mapSegmentVectors = over each

-- Note, can't yet have Haddock comments on GADT constructors; see
-- http://trac.haskell.org/haddock/ticket/43. For now we don't need
-- Segment to be a GADT but we might in the future. (?)

type instance V (Segment c v n) = v
type instance N (Segment c v n) = n

instance Transformable (Segment c v n) where
  transform = mapSegmentVectors . apply

instance Renderable (Segment c v n) NullBackend where
  render _ _ = mempty

-- | @'straight' v@ constructs a translationally invariant linear
--   segment with direction and length given by the vector @v@.
straight :: v n -> Segment Closed v n
straight = Linear . OffsetClosed

-- Note, if we didn't have a Linear constructor we could also create
-- linear segments with @Cubic (v ^/ 3) (2 *^ (v ^/ 3)) v@.  Those
-- would not be precisely the same, however, since we can actually
-- observe how segments are parametrized.

-- | @bezier3 c1 c2 x@ constructs a translationally invariant cubic
--   Bézier curve where the offsets from the first endpoint to the
--   first and second control point and endpoint are respectively
--   given by @c1@, @c2@, and @x@.
bezier3 :: v n -> v n -> v n -> Segment Closed v n
bezier3 c1 c2 x = Cubic c1 c2 (OffsetClosed x)

-- | @bézier3@ is the same as @bezier3@, but with more snobbery.
bézier3 :: v n -> v n -> v n -> Segment Closed v n
bézier3 = bezier3

type instance Codomain (Segment Closed v n) = v

-- | 'atParam' yields a parametrized view of segments as continuous
--   functions @[0,1] -> v@, which give the offset from the start of
--   the segment for each value of the parameter between @0@ and @1@.
--   It is designed to be used infix, like @seg ``atParam`` 0.5@.
instance (Additive v, Num n) => Parametric (Segment Closed v n) where
  atParam (Linear (OffsetClosed x)) t       = t *^ x
  atParam (Cubic c1 c2 (OffsetClosed x2)) t =     (3 * t'*t'*t ) *^ c1
                                              ^+^ (3 * t'*t *t ) *^ c2
                                              ^+^ (    t *t *t ) *^ x2
    where t' = 1-t

instance Num n => DomainBounds (Segment Closed v n)

instance (Additive v, Num n) => EndValues (Segment Closed v n) where
  atStart                            = const zero
  atEnd (Linear (OffsetClosed v))    = v
  atEnd (Cubic _ _ (OffsetClosed v)) = v

-- | Compute the offset from the start of a segment to the
--   end.  Note that in the case of a Bézier segment this is /not/ the
--   same as the length of the curve itself; for that, see 'arcLength'.
segOffset :: Segment Closed v n -> v n
segOffset (Linear (OffsetClosed v))    = v
segOffset (Cubic _ _ (OffsetClosed v)) = v

-- | An open linear segment. This means the trail makes a straight line
-- from the last segment the beginning to form a loop.
openLinear :: Segment Open v n
openLinear = Linear OffsetOpen

-- | An open cubic segment. This means the trail makes a cubic bézier
-- with control vectors @v1@ and @v2@ to form a loop.
openCubic :: v n -> v n -> Segment Open v n
openCubic v1 v2 = Cubic v1 v2 OffsetOpen

------------------------------------------------------------
--  Computing segment envelope  ------------------------------
------------------------------------------------------------

{- 3 (1-t)^2 t c1 + 3 (1-t) t^2 c2 + t^3 x2

   Can we compute the projection of B(t) onto a given vector v?

   u.v = |u||v| cos th

   |proj_v u| = cos th * |u|
              = (u.v/|v|)

   so B_v(t) = (B(t).v/|v|)

   Then take the derivative of this wrt. t, get a quadratic, solve.

   B_v(t) = (1/|v|) *     -- note this does not affect max/min, can solve for t first
            3 (1-t)^2 t (c1.v) + 3 (1-t) t^2 (c2.v) + t^3 (x2.v)
          = t^3 ((3c1 - 3c2 + x2).v) + t^2 ((-6c1 + 3c2).v) + t (3c1.v)

   B_v'(t) = t^2 (3(3c1 - 3c2 + x2).v) + t (6(-2c1 + c2).v) + 3c1.v

   Set equal to zero, use quadratic formula.
-}

-- | The envelope for a segment is based at the segment's start.
instance (Metric v, OrderedField n) => Enveloped (Segment Closed v n) where

  getEnvelope (s@(Linear {})) = mkEnvelope $ \v ->
    maximum (map (\t -> (s `atParam` t) `dot` v) [0,1]) / quadrance v

  getEnvelope (s@(Cubic c1 c2 (OffsetClosed x2))) = mkEnvelope $ \v ->
    maximum .
    map (\t -> ((s `atParam` t) `dot` v) / quadrance v) $
    [0,1] ++
    filter (liftA2 (&&) (>0) (<1))
      (quadForm (3 * ((3 *^ c1 ^-^ 3 *^ c2 ^+^ x2) `dot` v))
                (6 * (((-2) *^ c1 ^+^ c2) `dot` v))
                ((3 *^ c1) `dot` v))

------------------------------------------------------------
--  Manipulating segments
------------------------------------------------------------

instance (Additive v, Fractional n) => Sectionable (Segment Closed v n) where
  splitAtParam (Linear (OffsetClosed x1)) t = (left, right)
    where left  = straight p
          right = straight (x1 ^-^ p)
          p = lerp t zero x1
  splitAtParam (Cubic c1 c2 (OffsetClosed x2)) t = (left, right)
    where left  = bezier3 a b e
          right = bezier3 (c ^-^ e) (d ^-^ e) (x2 ^-^ e)
          p = lerp t c1 c2
          a = lerp t zero c1
          b = lerp t a p
          d = lerp t c2 x2
          c = lerp t p d
          e = lerp t b c

  reverseDomain = reverseSegment

-- | Reverse the direction of a segment.
reverseSegment :: (Num n, Additive v) => Segment Closed v n -> Segment Closed v n
reverseSegment (Linear (OffsetClosed v))       = straight (negated v)
reverseSegment (Cubic c1 c2 (OffsetClosed x2)) = bezier3 (c2 ^-^ x2) (c1 ^-^ x2) (negated x2)

-- Imitates I.elem for intervals<0.8 and I.member for intervals>=0.8
member :: Ord a => a -> I.Interval a -> Bool
member x (I.I a b) = x >= a && x <= b
{-# INLINE member #-}

instance (Metric v, OrderedField n)
      => HasArcLength (Segment Closed v n) where

  arcLengthBounded _ (Linear (OffsetClosed x1)) = I.singleton $ norm x1
  arcLengthBounded m s@(Cubic c1 c2 (OffsetClosed x2))
    | ub - lb < m = I lb ub
    | otherwise   = arcLengthBounded (m/2) l + arcLengthBounded (m/2) r
   where (l,r) = s `splitAtParam` 0.5
         ub    = sum (map norm [c1, c2 ^-^ c1, x2 ^-^ c2])
         lb    = norm x2

  arcLengthToParam m s _ | arcLength m s == 0 = 0.5
  arcLengthToParam m s@(Linear {}) len = len / arcLength m s
  arcLengthToParam m s@(Cubic {})  len
    | len `member` I (-m/2) (m/2) = 0
    | len < 0              = - arcLengthToParam m (fst (splitAtParam s (-1))) (-len)
    | len `member` slen    = 1
    | len > I.sup slen     = 2 * arcLengthToParam m (fst (splitAtParam s 2)) len
    | len < I.sup llen     = (*0.5) $ arcLengthToParam m l len
    | otherwise            = (+0.5) . (*0.5)
                           $ arcLengthToParam (9*m/10) r (len - I.midpoint llen)
    where (l,r) = s `splitAtParam` 0.5
          llen  = arcLengthBounded (m/10) l
          slen  = arcLengthBounded m s

  -- Note, the above seems to be quite slow since it duplicates a lot of
  -- work.  We could trade off some time for space by building a tree of
  -- parameter values (up to a certain depth...)

------------------------------------------------------------
--  Fixed segments
------------------------------------------------------------

-- | @FixedSegment@s are like 'Segment's except that they have
--   absolute locations.  @FixedSegment v@ is isomorphic to @Located
--   (Segment Closed v)@, as witnessed by 'mkFixedSeg' and
--   'fromFixedSeg', but @FixedSegment@ is convenient when one needs
--   the absolute locations of the vertices and control points.
data FixedSegment v n = FLinear (Point v n) (Point v n)
                      | FCubic (Point v n) (Point v n) (Point v n) (Point v n)
  deriving (Eq, Ord, Show)

type instance V (FixedSegment v n) = v
type instance N (FixedSegment v n) = n

instance Each (FixedSegment v n) (FixedSegment v' n') (Point v n) (Point v' n') where
  each f (FLinear p0 p1)      = FLinear <$> f p0 <*> f p1
  each f (FCubic p0 p1 p2 p3) = FCubic  <$> f p0 <*> f p1 <*> f p2 <*> f p3
  {-# INLINE each #-}

-- | Reverses the control points.
instance Reversing (FixedSegment v n) where
  reversing (FLinear p0 p1)      = FLinear p1 p0
  reversing (FCubic p0 p1 p2 p3) = FCubic p3 p2 p1 p0

instance (Additive v, Num n) => Transformable (FixedSegment v n) where
  transform t = over each (papply t)

instance (Additive v, Num n) => HasOrigin (FixedSegment v n) where
  moveOriginTo o = over each (moveOriginTo o)

instance (Metric v, OrderedField n) => Enveloped (FixedSegment v n) where
  getEnvelope f = moveTo p (getEnvelope s)
    where (p, s) = viewLoc $ fromFixedSeg f

    -- Eventually we might decide it's cleaner/more efficient (?) to
    -- have all the computation in the FixedSegment instance of
    -- Envelope, and implement the Segment instance in terms of it,
    -- instead of the other way around

instance (Metric v, OrderedField n)
      => HasArcLength (FixedSegment v n) where
  arcLengthBounded m s = arcLengthBounded m (fromFixedSeg s)
  arcLengthToParam m s = arcLengthToParam m (fromFixedSeg s)

-- | Create a 'FixedSegment' from a located 'Segment'.
mkFixedSeg :: (Num n, Additive v) => Located (Segment Closed v n) -> FixedSegment v n
mkFixedSeg ls =
  case viewLoc ls of
    (p, Linear (OffsetClosed v))       -> FLinear p (p .+^ v)
    (p, Cubic c1 c2 (OffsetClosed x2)) -> FCubic  p (p .+^ c1) (p .+^ c2) (p .+^ x2)

-- | Convert a 'FixedSegment' back into a located 'Segment'.
fromFixedSeg :: (Num n, Additive v) => FixedSegment v n -> Located (Segment Closed v n)
fromFixedSeg (FLinear p1 p2)      = straight (p2 .-. p1) `at` p1
fromFixedSeg (FCubic x1 c1 c2 x2) = bezier3 (c1 .-. x1) (c2 .-. x1) (x2 .-. x1) `at` x1

-- | Use a 'FixedSegment' to make an 'Iso' between an
-- a fixed segment and a located segment.
fixedSegIso :: (Num n, Additive v) => Iso' (FixedSegment v n) (Located (Segment Closed v n))
fixedSegIso = iso fromFixedSeg mkFixedSeg

type instance Codomain (FixedSegment v n) = Point v

instance (Additive v, Num n) => Parametric (FixedSegment v n) where
  atParam (FLinear p1 p2) t = lerp t p1 p2
  atParam (FCubic x1 c1 c2 x2) t = p3
    where p11 = lerp t x1 c1
          p12 = lerp t c1 c2
          p13 = lerp t c2 x2

          p21 = lerp t p11 p12
          p22 = lerp t p12 p13

          p3  = lerp t p21 p22

instance Num n => DomainBounds (FixedSegment v n)

instance (Additive v, Num n) => EndValues (FixedSegment v n) where
  atStart (FLinear p0 _)     = p0
  atStart (FCubic  p0 _ _ _) = p0
  atEnd   (FLinear _ p1)     = p1
  atEnd   (FCubic _ _ _ p1 ) = p1

instance (Additive v, Fractional n) => Sectionable (FixedSegment v n) where
  splitAtParam (FLinear p0 p1) t = (left, right)
    where left  = FLinear p0 p
          right = FLinear p  p1
          p = lerp t p0 p1
  splitAtParam (FCubic p0 c1 c2 p1) t = (left, right)
    where left  = FCubic p0 a b cut
          right = FCubic cut c d p1
          -- first round
          a   = lerp t p0 c1
          p   = lerp t c1 c2
          d   = lerp t c2 p1
          -- second round
          b   = lerp t a p
          c   = lerp t p d
          -- final round
          cut = lerp t b c

  reverseDomain (FLinear p0 p1) = FLinear p1 p0
  reverseDomain (FCubic p0 c1 c2 p1) = FCubic p1 c2 c1 p0

------------------------------------------------------------
--  Segment measures  --------------------------------------
------------------------------------------------------------

-- $segmeas
-- Trails store a sequence of segments in a fingertree, which can
-- automatically track various monoidal \"measures\" on segments.

-- | A type to track the count of segments in a 'Trail'.
newtype SegCount = SegCount (Sum Int)
  deriving (Semigroup, Monoid)

instance Wrapped SegCount where
  type Unwrapped SegCount = Sum Int
  _Wrapped' = iso (\(SegCount x) -> x) SegCount

instance Rewrapped SegCount SegCount

-- | A type to represent the total arc length of a chain of
--   segments. The first component is a \"standard\" arc length,
--   computed to within a tolerance of @10e-6@.  The second component is
--   a generic arc length function taking the tolerance as an
--   argument.

newtype ArcLength n
  = ArcLength (Sum (Interval n), n -> Sum (Interval n))

instance Wrapped (ArcLength n) where
  type Unwrapped (ArcLength n) = (Sum (Interval n), n -> Sum (Interval n))
  _Wrapped' = iso (\(ArcLength x) -> x) ArcLength

instance Rewrapped (ArcLength n) (ArcLength n')

-- | Project out the cached arc length, stored together with error
--   bounds.
getArcLengthCached :: ArcLength n -> Interval n
getArcLengthCached = getSum . fst . op ArcLength

-- | Project out the generic arc length function taking the tolerance as
--   an argument.
getArcLengthFun :: ArcLength n -> n -> Interval n
getArcLengthFun = fmap getSum . snd . op ArcLength

-- | Given a specified tolerance, project out the cached arc length if
--   it is accurate enough; otherwise call the generic arc length
--   function with the given tolerance.
getArcLengthBounded :: (Num n, Ord n)
                    => n -> ArcLength n -> Interval n
getArcLengthBounded eps al
  | I.width cached <= eps = cached
  | otherwise             = getArcLengthFun al eps
  where
    cached = getArcLengthCached al
deriving instance (Num n, Ord n) => Semigroup (ArcLength n)
deriving instance (Num n, Ord n) => Monoid    (ArcLength n)

-- | A type to represent the total cumulative offset of a chain of
--   segments.
newtype TotalOffset v n = TotalOffset (v n)

instance Wrapped (TotalOffset v n) where
  type Unwrapped (TotalOffset v n) = v n
  _Wrapped' = iso (\(TotalOffset x) -> x) TotalOffset

instance Rewrapped (TotalOffset v n) (TotalOffset v' n')

instance (Num n, Additive v) => Semigroup (TotalOffset v n) where
  TotalOffset v1 <> TotalOffset v2 = TotalOffset (v1 ^+^ v2)

instance (Num n, Additive v) => Monoid (TotalOffset v n) where
  mempty  = TotalOffset zero
  mappend = (<>)

-- | A type to represent the offset and envelope of a chain of
--   segments.  They have to be paired into one data structure, since
--   combining the envelopes of two consecutive chains needs to take
--   the offset of the first into account.
data OffsetEnvelope v n = OffsetEnvelope
  { _oeOffset   :: !(TotalOffset v n)
  , _oeEnvelope :: Envelope v n
  }

makeLenses ''OffsetEnvelope

instance (Metric v, OrderedField n) => Semigroup (OffsetEnvelope v n) where
  (OffsetEnvelope o1 e1) <> (OffsetEnvelope o2 e2)
    = let !negOff = negated . op TotalOffset $ o1
          e2Off = moveOriginBy negOff e2
          !_unused = maybe () (\f -> f `seq` ()) $ appEnvelope e2Off
      in OffsetEnvelope
          (o1 <> o2)
          (e1 <> e2Off)

-- | @SegMeasure@ collects up all the measurements over a chain of
--   segments.
type SegMeasure v n = SegCount
                  ::: ArcLength n
                  ::: OffsetEnvelope v n
                  ::: ()
  -- unfortunately we can't cache Trace, since there is not a generic
  -- instance Traced (Segment Closed v), only Traced (Segment Closed R2).

instance (Metric v, OrderedField n)
    => Measured (SegMeasure v n) (SegMeasure v n) where
  measure = id

instance (OrderedField n, Metric v)
    => Measured (SegMeasure v n) (Segment Closed v n) where
  measure s = (SegCount . Sum) 1

            -- cache arc length with two orders of magnitude more
            -- accuracy than standard, so we have a hope of coming out
            -- with an accurate enough total arc length for
            -- reasonable-length trails
            *: ArcLength ( Sum $ arcLengthBounded (stdTolerance/100) s
                         , Sum . flip arcLengthBounded s               )

            *: OffsetEnvelope (TotalOffset . segOffset $ s)
                              (getEnvelope s)

            *: ()

------------------------------------------------------------
--  Serialize instances
------------------------------------------------------------

instance (Serialize (v n)) => Serialize (Segment Open v n) where
  {-# INLINE put #-}
  put segment = case segment of
    Linear OffsetOpen    -> Serialize.put True
    Cubic v w OffsetOpen -> do
      Serialize.put False
      Serialize.put v
      Serialize.put w

  {-# INLINE get #-}
  get = do
    isLinear <- Serialize.get
    case isLinear of
      True  -> return (Linear OffsetOpen)
      False -> do
        v <- Serialize.get
        w <- Serialize.get
        return (Cubic v w OffsetOpen)

instance (Serialize (v n)) => Serialize (Segment Closed v n) where
  {-# INLINE put #-}
  put segment = case segment of
    Linear (OffsetClosed z)    -> do
      Serialize.put z
      Serialize.put True
    Cubic v w (OffsetClosed z) -> do
      Serialize.put z
      Serialize.put False
      Serialize.put v
      Serialize.put w

  {-# INLINE get #-}
  get = do
    z <- Serialize.get
    isLinear <- Serialize.get
    case isLinear of
      True  -> return (Linear (OffsetClosed z))
      False -> do
        v <- Serialize.get
        w <- Serialize.get
        return (Cubic v w (OffsetClosed z))
