diagrams-lib-1.5.1: src/Diagrams/Segment.hs
{-# 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))