{-# LANGUAGE EmptyDataDecls #-} {-# LANGUAGE FlexibleContexts #-} {-# LANGUAGE FlexibleInstances #-} {-# LANGUAGE GADTs #-} {-# LANGUAGE GeneralizedNewtypeDeriving #-} {-# LANGUAGE LambdaCase #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE StandaloneDeriving #-} {-# LANGUAGE TypeFamilies #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE UndecidableInstances #-} {-# LANGUAGE ViewPatterns #-} {-# OPTIONS_GHC -fno-warn-orphans #-} {-# OPTIONS_GHC -fno-warn-name-shadowing #-} -- We have an orphan Transformable FingerTree instance here. ----------------------------------------------------------------------------- -- | -- Module : Diagrams.Trail -- Copyright : (c) 2013-2015 diagrams-lib team (see LICENSE) -- License : BSD-style (see LICENSE) -- Maintainer : diagrams-discuss@googlegroups.com -- -- This module defines /trails/, translationally invariant paths -- through space. Trails form a central part of the diagrams-lib API, -- so the documentation for this module merits careful study. -- -- Related modules include: -- -- * The 'TrailLike' class ("Diagrams.TrailLike") exposes a generic -- API for building a wide range of things out of trails. -- -- * 'Path's ("Diagrams.Path") are collections of 'Located' -- ("Diagrams.Located") trails. -- -- * Trails are composed of 'Segment's (see "Diagrams.Segment"), -- though most users should not need to work with segments directly. -- ----------------------------------------------------------------------------- module Diagrams.Trail ( -- * Type definitions -- ** Lines and loops Trail'(..) , glueLine , closeLine , cutLoop -- ** Generic trails , Trail(..) , _Line, _Loop , wrapTrail, wrapLine, wrapLoop , onTrail, onLine , glueTrail, closeTrail, cutTrail -- * Constructing trails , emptyLine, emptyTrail , lineFromVertices, trailFromVertices , lineFromOffsets, trailFromOffsets , lineFromSegments, trailFromSegments , loopFromSegments -- * Eliminating trails , withTrail', withTrail, withLine , isLineEmpty, isTrailEmpty , isLine, isLoop , trailSegments, lineSegments, loopSegments , onLineSegments , trailOffsets, trailOffset , lineOffsets, lineOffset, loopOffsets , trailPoints, linePoints, loopPoints , trailVertices', lineVertices', loopVertices' , trailVertices, lineVertices, loopVertices , trailLocSegments, fixTrail -- * Modifying trails , reverseTrail, reverseLocTrail , reverseLine, reverseLocLine , reverseLoop, reverseLocLoop -- * Internals -- $internals -- ** Type tags , Line, Loop -- ** Segment trees , SegTree(..), trailMeasure, numSegs, offset -- ** Extracting segments , GetSegment(..), getSegment ) where import Control.Arrow ((***)) import Control.Lens hiding (at, transform, (<|), (|>)) import Data.FingerTree (FingerTree, ViewL (..), ViewR (..), (<|), (|>)) import qualified Data.FingerTree as FT import Data.Fixed import qualified Data.Foldable as F import Data.Monoid.MList import Data.Semigroup import qualified Numeric.Interval.Kaucher as I import Diagrams.Core import Diagrams.Located import Diagrams.Parametric import Diagrams.Segment import Diagrams.Tangent import Linear.Affine import Linear.Metric import Linear.Vector -- $internals -- -- Most users of diagrams should not need to use anything in this -- section directly, but they are exported on the principle that we -- can't forsee what uses people might have for them. ------------------------------------------------------------ -- FingerTree instances ------------------------------------------------------------ type instance V (FingerTree m a) = V a type instance N (FingerTree m a) = N a instance (FT.Measured m a, Transformable a) => Transformable (FingerTree m a) where transform = FT.fmap' . transform instance (FT.Measured m a, FT.Measured n b) => Cons (FingerTree m a) (FingerTree n b) a b where _Cons = prism (uncurry (FT.<|)) $ \aas -> case FT.viewl aas of a FT.:< as -> Right (a, as) EmptyL -> Left mempty {-# INLINE _Cons #-} instance (FT.Measured m a, FT.Measured n b) => Snoc (FingerTree m a) (FingerTree n b) a b where _Snoc = prism (uncurry (FT.|>)) $ \aas -> case FT.viewr aas of as FT.:> a -> Right (as, a) EmptyR -> Left mempty {-# INLINE _Snoc #-} ------------------------------------------------------------ -- Segment trees ----------------------------------------- ------------------------------------------------------------ -- | A @SegTree@ represents a sequence of closed segments, stored in a -- fingertree so we can easily recover various monoidal measures of -- the segments (number of segments, arc length, envelope...) and -- also easily slice and dice them according to the measures -- (/e.g./, split off the smallest number of segments from the -- beginning which have a combined arc length of at least 5). newtype SegTree v n = SegTree (FingerTree (SegMeasure v n) (Segment Closed v n)) deriving (Eq, Ord, Show, Monoid, Transformable, FT.Measured (SegMeasure v n)) instance Wrapped (SegTree v n) where type Unwrapped (SegTree v n) = FingerTree (SegMeasure v n) (Segment Closed v n) _Wrapped' = iso (\(SegTree x) -> x) SegTree {-# INLINE _Wrapped' #-} instance (Metric v, OrderedField n, Metric u, OrderedField n') => Cons (SegTree v n) (SegTree u n') (Segment Closed v n) (Segment Closed u n') where _Cons = _Wrapped . _Cons . bimapping id _Unwrapped {-# INLINE _Cons #-} instance (Metric v, OrderedField n, Metric u, OrderedField n') => Snoc (SegTree v n) (SegTree u n') (Segment Closed v n) (Segment Closed u n') where _Snoc = _Wrapped . _Snoc . bimapping _Unwrapped id {-# INLINE _Snoc #-} instance Rewrapped (SegTree v n) (SegTree v' n') type instance V (SegTree v n) = v type instance N (SegTree v n) = n type instance Codomain (SegTree v n) = v instance (Metric v, OrderedField n, Real n) => Parametric (SegTree v n) where atParam t p = offset . fst $ splitAtParam t p instance Num n => DomainBounds (SegTree v n) instance (Metric v, OrderedField n, Real n) => EndValues (SegTree v n) type SplitResult v n = ((SegTree v n, n -> n), (SegTree v n, n -> n)) splitAtParam' :: (Metric v, OrderedField n, Real n) => SegTree v n -> n -> SplitResult v n splitAtParam' tree@(SegTree t) p | p < 0 = case FT.viewl t of EmptyL -> emptySplit seg FT.:< t' -> case seg `splitAtParam` (p * tSegs) of (seg1, seg2) -> ( (SegTree $ FT.singleton seg1, (*p)) , (SegTree $ seg2 <| t', \u -> 1 - (1 - u) * tSegs / (tSegs + 1)) ) | p >= 1 = case FT.viewr t of EmptyR -> emptySplit t' FT.:> seg -> case seg `splitAtParam` (1 - (1 - p)*tSegs) of (seg1, seg2) -> ( (SegTree $ t' |> seg1, \u -> u * tSegs / (tSegs + 1)) , (SegTree $ FT.singleton seg2, \u -> (u - p) / (1 - p)) ) | otherwise = case FT.viewl after of EmptyL -> emptySplit seg FT.:< after' -> let (n, p') = propFrac $ p * tSegs f p n u | u * tSegs < n = u * tSegs / (n + 1) | otherwise = (n + (u * tSegs - n) / (p * tSegs - n)) / (n+1) in case seg `splitAtParam` p' of (seg1, seg2) -> ( ( SegTree $ before |> seg1 , f p n ) , ( SegTree $ seg2 <| after' , \v -> 1 - f (1 - p) (tSegs - n - 1) (1 - v) ) ) where (before, after) = FT.split ((p * tSegs <) . numSegs) t tSegs = numSegs t emptySplit = let t' = (tree, id) in (t',t') propFrac x = let m = signum x * mod1 x in (x - m, m) instance (Metric v, OrderedField n, Real n) => Sectionable (SegTree v n) where splitAtParam tree p = let ((a,_),(b,_)) = splitAtParam' tree p in (a,b) reverseDomain (SegTree t) = SegTree $ FT.reverse t' where t' = FT.fmap' reverseSegment t section x t1 t2 = let ((a,fa),_) = splitAtParam' x t2 in snd $ splitAtParam a (fa t1) -- XXX seems like it should be possible to collapse some of the -- above cases into one? instance (Metric v, OrderedField n, Real n) => HasArcLength (SegTree v n) where arcLengthBounded eps t -- Use the cached value if it is accurate enough; otherwise fall -- back to recomputing a more accurate value | I.width i <= eps = i | otherwise = fun (eps / numSegs t) where i = trailMeasure (I.singleton 0) getArcLengthCached t fun = trailMeasure (const 0) getArcLengthFun t arcLengthToParam eps st@(SegTree t) l | l < 0 = case FT.viewl t of EmptyL -> 0 seg FT.:< _ -> arcLengthToParam eps seg l / tSegs | l >= totalAL = case FT.viewr t of EmptyR -> 0 t' FT.:> seg -> let p = arcLengthToParam (eps/2) seg (l - arcLength (eps/2) (SegTree t')) in (p - 1)/tSegs + 1 | otherwise = case FT.viewl after of EmptyL -> 0 seg FT.:< _ -> let p = arcLengthToParam (eps/2) seg (l - arcLength (eps/2) (SegTree before)) in (numSegs before + p) / tSegs where totalAL = arcLength eps st tSegs = numSegs t before, after :: FingerTree (SegMeasure v n) (Segment Closed v n) (before, after) = FT.split ((>= l) . trailMeasure 0 (I.midpoint . getArcLengthBounded eps)) t -- | Given a default result (to be used in the case of an empty -- trail), and a function to map a single measure to a result, -- extract the given measure for a trail and use it to compute a -- result. Put another way, lift a function on a single measure -- (along with a default value) to a function on an entire trail. trailMeasure :: ( SegMeasure v n :>: m, FT.Measured (SegMeasure v n) t ) => a -> (m -> a) -> t -> a trailMeasure d f = option d f . get . FT.measure -- | Compute the number of segments of anything measured by -- 'SegMeasure' (/e.g./ @SegMeasure@ itself, @Segment@, @SegTree@, -- @Trail@s...) numSegs :: (Num c, FT.Measured (SegMeasure v n) a) => a -> c numSegs = fromIntegral . trailMeasure 0 (getSum . op SegCount) -- | Compute the total offset of anything measured by 'SegMeasure'. offset :: ( OrderedField n, Metric v, FT.Measured (SegMeasure v n) t ) => t -> v n offset = trailMeasure zero (op TotalOffset . view oeOffset) ------------------------------------------------------------ -- Trails ------------------------------------------------ ------------------------------------------------------------ -- Eventually we should use DataKinds for this, but not until we drop -- support for GHC 7.4. -- | Type tag for trails with distinct endpoints. data Line -- | Type tag for \"loopy\" trails which return to their starting -- point. data Loop -------------------------------------------------- -- The Trail' type -- | Intuitively, a trail is a single, continuous path through space. -- However, a trail has no fixed starting point; it merely specifies -- /how/ to move through space, not /where/. For example, \"take -- three steps forward, then turn right twenty degrees and take two -- more steps\" is an intuitive analog of a trail; these -- instructions specify a path through space from any given starting -- location. To be precise, trails are /translation-invariant/; -- applying a translation to a trail has no effect. -- -- A @'Located' Trail@, on the other hand, is a trail paired with -- some concrete starting location (\"start at the big tree on the -- corner, then take three steps forward, ...\"). See the -- "Diagrams.Located" module for help working with 'Located' values. -- -- Formally, the semantics of a trail is a continuous (though not -- necessarily differentiable) function from the real interval [0,1] -- to vectors in some vector space. (In contrast, a 'Located' trail -- is a continuous function from [0,1] to /points/ in some /affine/ -- space.) -- -- There are two types of trails: -- -- * A \"line\" (think of the \"train\", \"subway\", or \"bus\" -- variety, rather than the \"straight\" variety...) is a trail -- with two distinct endpoints. Actually, a line can have the -- same start and end points, but it is still /drawn/ as if it had -- distinct endpoints: the two endpoints will have the appropriate -- end caps, and the trail will not be filled. Lines have a -- @Monoid@ instance where @mappend@ corresponds to concatenation, -- /i.e./ chaining one line after the other. -- -- * A \"loop\" is required to end in the same place it starts (that -- is, t(0) = t(1)). Loops are filled and are drawn as one -- continuous loop, with the appropriate join at the -- start/endpoint rather than end caps. Loops do not have a -- @Monoid@ instance. -- -- To convert between lines and loops, see 'glueLine', -- 'closeLine', and 'cutLoop'. -- -- To construct trails, see 'emptyTrail', 'trailFromSegments', -- 'trailFromVertices', 'trailFromOffsets', and friends. You can -- also get any type of trail from any function which returns a -- 'TrailLike' (/e.g./ functions in "Diagrams.TwoD.Shapes", and many -- others; see "Diagrams.TrailLike"). -- -- To extract information from trails, see 'withLine', 'isLoop', -- 'trailSegments', 'trailOffsets', 'trailVertices', and friends. data Trail' l v n where Line :: SegTree v n -> Trail' Line v n Loop :: SegTree v n -> Segment Open v n -> Trail' Loop v n -- | A generic eliminator for 'Trail'', taking functions specifying -- what to do in the case of a line or a loop. withTrail' :: (Trail' Line v n -> r) -> (Trail' Loop v n -> r) -> Trail' l v n -> r withTrail' line _ t@(Line{}) = line t withTrail' _ loop t@(Loop{}) = loop t deriving instance Eq (v n) => Eq (Trail' l v n) deriving instance Ord (v n) => Ord (Trail' l v n) instance Show (v n) => Show (Trail' l v n) where showsPrec d (Line (SegTree ft)) = showParen (d > 10) $ showString "lineFromSegments " . showList (F.toList ft) showsPrec d (Loop (SegTree ft) o) = showParen (d > 10) $ showString "loopFromSegments " . showList (F.toList ft) . showChar ' ' . showsPrec 11 o type instance V (Trail' l v n) = v type instance N (Trail' l v n) = n type instance Codomain (Trail' l v n) = v instance (OrderedField n, Metric v) => Semigroup (Trail' Line v n) where (Line t1) <> (Line t2) = Line (t1 `mappend` t2) -- | The empty trail is constantly the zero vector. Trails are -- composed via concatenation. Note that only lines have a monoid -- instance (and not loops). instance (Metric v, OrderedField n) => Monoid (Trail' Line v n) where mempty = emptyLine mappend = (<>) instance (Metric v, OrderedField n) => AsEmpty (Trail' Line v n) where _Empty = nearly emptyLine isLineEmpty instance (HasLinearMap v, Metric v, OrderedField n) => Transformable (Trail' l v n) where transform tr (Line t ) = Line (transform tr t) transform tr (Loop t s) = Loop (transform tr t) (transform tr s) -- | The envelope for a trail is based at the trail's start. instance (Metric v, OrderedField n) => Enveloped (Trail' l v n) where getEnvelope = withTrail' ftEnv (ftEnv . cutLoop) where ftEnv :: Trail' Line v n -> Envelope v n ftEnv (Line t) = trailMeasure mempty (view oeEnvelope) t instance (HasLinearMap v, Metric v, OrderedField n) => Renderable (Trail' o v n) NullBackend where render _ _ = mempty instance (Metric v, OrderedField n, Real n) => Parametric (Trail' l v n) where atParam t p = withTrail' (\(Line segT) -> segT `atParam` p) (\l -> cutLoop l `atParam` mod1 p) t instance (Parametric (GetSegment (Trail' c v n)), Additive v, Num n) => Parametric (Tangent (Trail' c v n)) where Tangent tr `atParam` p = case GetSegment tr `atParam` p of GetSegmentCodomain Nothing -> zero GetSegmentCodomain (Just (_, seg, reparam)) -> Tangent seg `atParam` (p ^. cloneIso reparam) instance ( Parametric (GetSegment (Trail' c v n)) , EndValues (GetSegment (Trail' c v n)) , Additive v , Num n ) => EndValues (Tangent (Trail' c v n)) where atStart (Tangent tr) = case atStart (GetSegment tr) of GetSegmentCodomain Nothing -> zero GetSegmentCodomain (Just (_, seg, _)) -> atStart (Tangent seg) atEnd (Tangent tr) = case atEnd (GetSegment tr) of GetSegmentCodomain Nothing -> zero GetSegmentCodomain (Just (_, seg, _)) -> atEnd (Tangent seg) instance (Metric v , OrderedField n, Real n) => Parametric (Tangent (Trail v n)) where Tangent tr `atParam` p = withTrail ((`atParam` p) . Tangent) ((`atParam` p) . Tangent) tr instance (Metric v, OrderedField n, Real n) => EndValues (Tangent (Trail v n)) where atStart (Tangent tr) = withTrail (atStart . Tangent) (atStart . Tangent) tr atEnd (Tangent tr) = withTrail (atEnd . Tangent) (atEnd . Tangent) tr -- | Compute the remainder mod 1. Convenient for constructing loop -- parameterizations that wrap around. mod1 :: Real a => a -> a mod1 = (`mod'` 1) instance Num n => DomainBounds (Trail' l v n) instance (Metric v, OrderedField n, Real n) => EndValues (Trail' l v n) instance (Metric v, OrderedField n, Real n) => Sectionable (Trail' Line v n) where splitAtParam (Line t) p = (Line t1, Line t2) where (t1, t2) = splitAtParam t p reverseDomain = reverseLine instance (Metric v, OrderedField n, Real n) => HasArcLength (Trail' l v n) where arcLengthBounded eps = withTrail' (\(Line t) -> arcLengthBounded eps t) (arcLengthBounded eps . cutLoop) arcLengthToParam eps tr l = withTrail' (\(Line t) -> arcLengthToParam eps t l) (\lp -> arcLengthToParam eps (cutLoop lp) l) tr instance Rewrapped (Trail' Line v n) (Trail' Line v' n') instance Wrapped (Trail' Line v n) where type Unwrapped (Trail' Line v n) = SegTree v n _Wrapped' = iso (\(Line x) -> x) Line {-# INLINE _Wrapped' #-} instance (Metric v, OrderedField n, Metric u, OrderedField n') => Cons (Trail' Line v n) (Trail' Line u n') (Segment Closed v n) (Segment Closed u n') where _Cons = _Wrapped . _Cons . bimapping id _Unwrapped {-# INLINE _Cons #-} instance (Metric v, OrderedField n, Metric u, OrderedField n') => Snoc (Trail' Line v n) (Trail' Line u n') (Segment Closed v n) (Segment Closed u n') where _Snoc = _Wrapped . _Snoc . bimapping _Unwrapped id {-# INLINE _Snoc #-} -------------------------------------------------- -- Extracting segments -- | A newtype wrapper around trails which exists solely for its -- 'Parametric', 'DomainBounds' and 'EndValues' instances. The idea -- is that if @tr@ is a trail, you can write, /e.g./ -- -- @ -- getSegment tr `atParam` 0.6 -- @ -- -- or -- -- @ -- atStart (getSegment tr) -- @ -- -- to get the segment at parameter 0.6 or the first segment in the -- trail, respectively. -- -- The codomain for 'GetSegment', /i.e./ the result you get from -- calling 'atParam', 'atStart', or 'atEnd', is @Maybe (v, Segment -- Closed v, AnIso' n n)@. @Nothing@ results if -- the trail is empty; otherwise, you get: -- -- * the offset from the start of the trail to the beginning of the -- segment, -- -- * the segment itself, and -- -- * a reparameterization isomorphism: in the forward direction, it -- translates from parameters on the whole trail to a parameters -- on the segment. Note that for technical reasons you have to -- call 'cloneIso' on the @AnIso'@ value to get a real isomorphism -- you can use. newtype GetSegment t = GetSegment t newtype GetSegmentCodomain v n = GetSegmentCodomain (Maybe ( v n -- offset from trail start to segment start , Segment Closed v n -- the segment , AnIso' n n -- reparameterization, trail <-> segment )) -- | Create a 'GetSegment' wrapper around a trail, after which you can -- call 'atParam', 'atStart', or 'atEnd' to extract a segment. getSegment :: t -> GetSegment t getSegment = GetSegment type instance V (GetSegment t) = V t type instance N (GetSegment t) = N t type instance Codomain (GetSegment t) = GetSegmentCodomain (V t) -- | Parameters less than 0 yield the first segment; parameters -- greater than 1 yield the last. A parameter exactly at the -- junction of two segments yields the second segment (/i.e./ the -- one with higher parameter values). instance (Metric v, OrderedField n) => Parametric (GetSegment (Trail' Line v n)) where atParam (GetSegment (Line (SegTree ft))) p | p <= 0 = case FT.viewl ft of EmptyL -> GetSegmentCodomain Nothing seg FT.:< _ -> GetSegmentCodomain $ Just (zero, seg, reparam 0) | p >= 1 = case FT.viewr ft of EmptyR -> GetSegmentCodomain Nothing ft' FT.:> seg -> GetSegmentCodomain $ Just (offset ft', seg, reparam (n-1)) | otherwise = let (before, after) = FT.split ((p*n <) . numSegs) ft in case FT.viewl after of EmptyL -> GetSegmentCodomain Nothing seg FT.:< _ -> GetSegmentCodomain $ Just (offset before, seg, reparam (numSegs before)) where n = numSegs ft reparam k = iso (subtract k . (*n)) ((/n) . (+ k)) -- | The parameterization for loops wraps around, /i.e./ parameters -- are first reduced \"mod 1\". instance (Metric v, OrderedField n, Real n) => Parametric (GetSegment (Trail' Loop v n)) where atParam (GetSegment l) p = atParam (GetSegment (cutLoop l)) (mod1 p) instance (Metric v, OrderedField n, Real n) => Parametric (GetSegment (Trail v n)) where atParam (GetSegment t) p = withTrail ((`atParam` p) . GetSegment) ((`atParam` p) . GetSegment) t instance DomainBounds t => DomainBounds (GetSegment t) where domainLower (GetSegment t) = domainLower t domainUpper (GetSegment t) = domainUpper t instance (Metric v, OrderedField n) => EndValues (GetSegment (Trail' Line v n)) where atStart (GetSegment (Line (SegTree ft))) = case FT.viewl ft of EmptyL -> GetSegmentCodomain Nothing seg FT.:< _ -> let n = numSegs ft in GetSegmentCodomain $ Just (zero, seg, iso (*n) (/n)) atEnd (GetSegment (Line (SegTree ft))) = case FT.viewr ft of EmptyR -> GetSegmentCodomain Nothing ft' FT.:> seg -> let n = numSegs ft in GetSegmentCodomain $ Just (offset ft', seg, iso (subtract (n-1) . (*n)) ((/n) . (+ (n-1))) ) instance (Metric v, OrderedField n, Real n) => EndValues (GetSegment (Trail' Loop v n)) where atStart (GetSegment l) = atStart (GetSegment (cutLoop l)) atEnd (GetSegment l) = atEnd (GetSegment (cutLoop l)) instance (Metric v, OrderedField n, Real n) => EndValues (GetSegment (Trail v n)) where atStart (GetSegment t) = withTrail (atStart . GetSegment) (atStart . GetSegment) t atEnd (GetSegment t) = withTrail (atEnd . GetSegment) (atEnd . GetSegment) t -------------------------------------------------- -- The Trail type -- | @Trail@ is a wrapper around @Trail'@, hiding whether the -- underlying @Trail'@ is a line or loop (though which it is can be -- recovered; see /e.g./ 'withTrail'). data Trail v n where Trail :: Trail' l v n -> Trail v n deriving instance Show (v n) => Show (Trail v n) instance Eq (v n) => Eq (Trail v n) where t1 == t2 = withTrail (\ln1 -> withTrail (\ln2 -> ln1 == ln2) (const False) t2) (\lp1 -> withTrail (const False) (\lp2 -> lp1 == lp2) t2) t1 instance Ord (v n) => Ord (Trail v n) where compare t1 t2 = withTrail (\ln1 -> withTrail (compare ln1) (const LT) t2) (\lp1 -> withTrail (const GT) (compare lp1) t2) t1 -- | Two @Trail@s are combined by first ensuring they are both lines -- (using 'cutTrail' on loops) and then concatenating them. The -- result, in general, is a line. However, there is a special case -- for the empty line, which acts as the identity (so combining the -- empty line with a loop results in a loop). instance (OrderedField n, Metric v) => Semigroup (Trail v n) where (Trail (Line (SegTree ft))) <> t2 | FT.null ft = t2 t1 <> (Trail (Line (SegTree ft))) | FT.null ft = t1 t1 <> t2 = flip withLine t1 $ \l1 -> flip withLine t2 $ \l2 -> wrapLine (l1 <> l2) -- | @Trail@s are combined as described in the 'Semigroup' instance; -- the empty line is the identity element, with special cases so -- that combining the empty line with a loop results in the -- unchanged loop (in all other cases loops will be cut). Note that -- this does, in fact, satisfy the monoid laws, though it is a bit -- strange. Mostly it is provided for convenience, so one can work -- directly with @Trail@s instead of working with @Trail' Line@s and -- then wrapping. instance (Metric v, OrderedField n) => Monoid (Trail v n) where mempty = wrapLine emptyLine mappend = (<>) instance (Metric v, OrderedField n) => AsEmpty (Trail v n) where _Empty = nearly emptyTrail isTrailEmpty type instance V (Trail v n) = v type instance N (Trail v n) = n type instance Codomain (Trail v n) = v instance (HasLinearMap v, Metric v, OrderedField n) => Transformable (Trail v n) where transform t = onTrail (transform t) (transform t) instance (Metric v, OrderedField n) => Enveloped (Trail v n) where getEnvelope = withTrail getEnvelope getEnvelope instance (Metric v, OrderedField n, Real n) => Parametric (Trail v n) where atParam t p = withTrail (`atParam` p) (`atParam` p) t instance Num n => DomainBounds (Trail v n) instance (Metric v, OrderedField n, Real n) => EndValues (Trail v n) -- | Note that there is no @Sectionable@ instance for @Trail' Loop@, -- because it does not make sense (splitting a loop at a parameter -- results in a single line, not two loops). However, it's -- convenient to have a @Sectionable@ instance for @Trail@; if the -- @Trail@ contains a loop the loop will first be cut and then -- @splitAtParam@ called on the resulting line. This is -- semantically a bit silly, so please don't rely on it. (*E.g.* if -- this is really the behavior you want, consider first calling -- 'cutLoop' yourself.) instance (Metric v, OrderedField n, Real n) => Sectionable (Trail v n) where splitAtParam t p = withLine ((wrapLine *** wrapLine) . (`splitAtParam` p)) t reverseDomain = reverseTrail instance (Metric v, OrderedField n, Real n) => HasArcLength (Trail v n) where arcLengthBounded = withLine . arcLengthBounded arcLengthToParam eps tr al = withLine (\ln -> arcLengthToParam eps ln al) tr -- lens instrances ----------------------------------------------------- -- | Prism onto a 'Line'. _Line :: Prism' (Trail v n) (Trail' Line v n) _Line = _Wrapped' . _Left -- | Prism onto a 'Loop'. _Loop :: Prism' (Trail v n) (Trail' Loop v n) _Loop = _Wrapped' . _Right instance Rewrapped (Trail v n) (Trail v' n') instance Wrapped (Trail v n) where type Unwrapped (Trail v n) = Either (Trail' Line v n) (Trail' Loop v n) _Wrapped' = iso getTrail (either Trail Trail) where getTrail :: Trail v n -> Either (Trail' Line v n) (Trail' Loop v n) getTrail (Trail t@(Line {})) = Left t getTrail (Trail t@(Loop {})) = Right t -------------------------------------------------- -- Constructors and eliminators for Trail -- | A generic eliminator for 'Trail', taking functions specifying -- what to do in the case of a line or a loop. withTrail :: (Trail' Line v n -> r) -> (Trail' Loop v n -> r) -> Trail v n -> r withTrail line loop (Trail t) = withTrail' line loop t -- | Modify a @Trail@, specifying two separate transformations for the -- cases of a line or a loop. onTrail :: (Trail' Line v n -> Trail' l1 v n) -> (Trail' Loop v n -> Trail' l2 v n) -> Trail v n -> Trail v n onTrail o c = withTrail (wrapTrail . o) (wrapTrail . c) -- | An eliminator for @Trail@ based on eliminating lines: if the -- trail is a line, the given function is applied; if it is a loop, it -- is first converted to a line with 'cutLoop'. That is, -- -- @ -- withLine f === 'withTrail' f (f . 'cutLoop') -- @ withLine :: (Metric v, OrderedField n) => (Trail' Line v n -> r) -> Trail v n -> r withLine f = withTrail f (f . cutLoop) -- | Modify a @Trail@ by specifying a transformation on lines. If the -- trail is a line, the transformation will be applied directly. If -- it is a loop, it will first be cut using 'cutLoop', the -- transformation applied, and then glued back into a loop with -- 'glueLine'. That is, -- -- @ -- onLine f === onTrail f (glueLine . f . cutLoop) -- @ -- -- Note that there is no corresponding @onLoop@ function, because -- there is no nice way in general to convert a line into a loop, -- operate on it, and then convert back. onLine :: (Metric v, OrderedField n) => (Trail' Line v n -> Trail' Line v n) -> Trail v n -> Trail v n onLine f = onTrail f (glueLine . f . cutLoop) -- | Convert a 'Trail'' into a 'Trail', hiding the type-level -- distinction between lines and loops. wrapTrail :: Trail' l v n -> Trail v n wrapTrail = Trail -- | Convert a line into a 'Trail'. This is the same as 'wrapTrail', -- but with a more specific type, which can occasionally be -- convenient for fixing the type of a polymorphic expression. wrapLine :: Trail' Line v n -> Trail v n wrapLine = wrapTrail -- | Convert a loop into a 'Trail'. This is the same as 'wrapTrail', -- but with a more specific type, which can occasionally be -- convenient for fixing the type of a polymorphic expression. wrapLoop :: Trail' Loop v n -> Trail v n wrapLoop = wrapTrail ------------------------------------------------------------ -- Constructing trails ----------------------------------- ------------------------------------------------------------ -- | The empty line, which is the identity for concatenation of lines. emptyLine :: (Metric v, OrderedField n) => Trail' Line v n emptyLine = Line mempty -- | A wrapped variant of 'emptyLine'. emptyTrail :: (Metric v, OrderedField n) => Trail v n emptyTrail = wrapLine emptyLine -- | Construct a line from a list of closed segments. lineFromSegments :: (Metric v, OrderedField n) => [Segment Closed v n] -> Trail' Line v n lineFromSegments = Line . SegTree . FT.fromList -- | Construct a loop from a list of closed segments and an open segment -- that completes the loop. loopFromSegments :: (Metric v, OrderedField n) => [Segment Closed v n] -> Segment Open v n -> Trail' Loop v n loopFromSegments segs = Loop (SegTree (FT.fromList segs)) -- | @trailFromSegments === 'wrapTrail' . 'lineFromSegments'@, for -- conveniently constructing a @Trail@ instead of a @Trail'@. trailFromSegments :: (Metric v, OrderedField n) => [Segment Closed v n] -> Trail v n trailFromSegments = wrapTrail . lineFromSegments -- | Construct a line containing only linear segments from a list of -- vectors, where each vector represents the offset from one vertex -- to the next. See also 'fromOffsets'. -- -- <<diagrams/src_Diagrams_Trail_lineFromOffsetsEx.svg#diagram=lineFromOffsetsEx&width=300>> -- -- > import Diagrams.Coordinates -- > lineFromOffsetsEx = strokeLine $ lineFromOffsets [ 2 ^& 1, 2 ^& (-1), 2 ^& 0.5 ] lineFromOffsets :: (Metric v, OrderedField n) => [v n] -> Trail' Line v n lineFromOffsets = lineFromSegments . map straight -- | @trailFromOffsets === 'wrapTrail' . 'lineFromOffsets'@, for -- conveniently constructing a @Trail@ instead of a @Trail' Line@. trailFromOffsets :: (Metric v, OrderedField n) => [v n] -> Trail v n trailFromOffsets = wrapTrail . lineFromOffsets -- | Construct a line containing only linear segments from a list of -- vertices. Note that only the relative offsets between the -- vertices matters; the information about their absolute position -- will be discarded. That is, for all vectors @v@, -- -- @ -- lineFromVertices === lineFromVertices . 'translate' v -- @ -- -- If you want to retain the position information, you should -- instead use the more general 'fromVertices' function to -- construct, say, a @'Located' ('Trail'' 'Line' v)@ or a @'Located' -- ('Trail' v)@. -- -- <<diagrams/src_Diagrams_Trail_lineFromVerticesEx.svg#diagram=lineFromVerticesEx&width=300>> -- -- > import Diagrams.Coordinates -- > lineFromVerticesEx = pad 1.1 . centerXY . strokeLine -- > $ lineFromVertices [origin, 0 ^& 1, 1 ^& 2, 5 ^& 1] lineFromVertices :: (Metric v, OrderedField n) => [Point v n] -> Trail' Line v n lineFromVertices [] = emptyLine lineFromVertices [_] = emptyLine lineFromVertices ps = lineFromSegments . map straight $ zipWith (.-.) (tail ps) ps -- | @trailFromVertices === 'wrapTrail' . 'lineFromVertices'@, for -- conveniently constructing a @Trail@ instead of a @Trail' Line@. trailFromVertices :: (Metric v, OrderedField n) => [Point v n] -> Trail v n trailFromVertices = wrapTrail . lineFromVertices ------------------------------------------------------------ -- Converting between lines and loops -------------------- ------------------------------------------------------------ -- | Make a line into a loop by \"gluing\" the endpoint to the -- starting point. In particular, the offset of the final segment -- is modified so that it ends at the starting point of the entire -- trail. Typically, you would first construct a line which you -- know happens to end where it starts, and then call 'glueLine' to -- turn it into a loop. -- -- <<diagrams/src_Diagrams_Trail_glueLineEx.svg#diagram=glueLineEx&width=500>> -- -- > glueLineEx = pad 1.1 . hsep 1 -- > $ [almostClosed # strokeLine, almostClosed # glueLine # strokeLoop] -- > -- > almostClosed :: Trail' Line V2 Double -- > almostClosed = fromOffsets $ map r2 [(2, -1), (-3, -0.5), (-2, 1), (1, 0.5)] -- -- @glueLine@ is left inverse to 'cutLoop', that is, -- -- @ -- glueLine . cutLoop === id -- @ glueLine :: (Metric v, OrderedField n) => Trail' Line v n -> Trail' Loop v n glueLine (Line (SegTree t)) = case FT.viewr t of FT.EmptyR -> Loop mempty (Linear OffsetOpen) t' FT.:> Linear _ -> Loop (SegTree t') (Linear OffsetOpen) t' FT.:> Cubic c1 c2 _ -> Loop (SegTree t') (Cubic c1 c2 OffsetOpen) -- | @glueTrail@ is a variant of 'glueLine' which works on 'Trail's. -- It performs 'glueLine' on lines and is the identity on loops. glueTrail :: (Metric v, OrderedField n) => Trail v n -> Trail v n glueTrail = onTrail glueLine id -- | Make a line into a loop by adding a new linear segment from the -- line's end to its start. -- -- @closeLine@ does not have any particularly nice theoretical -- properties, but can be useful /e.g./ when you want to make a -- closed polygon out of a list of points where the initial point is -- not repeated at the end. To use 'glueLine', one would first have -- to duplicate the initial vertex, like -- -- @ -- 'glueLine' . 'lineFromVertices' $ ps ++ [head ps] -- @ -- -- Using @closeLine@, however, one can simply -- -- @ -- closeLine . lineFromVertices $ ps -- @ -- -- <<diagrams/src_Diagrams_Trail_closeLineEx.svg#diagram=closeLineEx&width=500>> -- -- > closeLineEx = pad 1.1 . centerXY . hcat' (with & sep .~ 1) -- > $ [almostClosed # strokeLine, almostClosed # closeLine # strokeLoop] closeLine :: Trail' Line v n -> Trail' Loop v n closeLine (Line t) = Loop t (Linear OffsetOpen) -- | @closeTrail@ is a variant of 'closeLine' for 'Trail', which -- performs 'closeLine' on lines and is the identity on loops. closeTrail :: Trail v n -> Trail v n closeTrail = onTrail closeLine id -- | Turn a loop into a line by \"cutting\" it at the common start/end -- point, resulting in a line which just happens to start and end at -- the same place. -- -- @cutLoop@ is right inverse to 'glueLine', that is, -- -- @ -- glueLine . cutLoop === id -- @ cutLoop :: forall v n. (Metric v, OrderedField n) => Trail' Loop v n -> Trail' Line v n cutLoop (Loop (SegTree t) c) = case (FT.null t, c) of (True, Linear OffsetOpen) -> emptyLine (_ , Linear OffsetOpen) -> Line (SegTree (t |> Linear off)) (_ , Cubic c1 c2 OffsetOpen) -> Line (SegTree (t |> Cubic c1 c2 off)) where offV :: v n offV = negated . trailMeasure zero (op TotalOffset .view oeOffset) $ t off = OffsetClosed offV -- | @cutTrail@ is a variant of 'cutLoop' for 'Trail'; it is the is -- the identity on lines and performs 'cutLoop' on loops. cutTrail :: (Metric v, OrderedField n) => Trail v n -> Trail v n cutTrail = onTrail id cutLoop ------------------------------------------------------------ -- Eliminating trails ------------------------------------ ------------------------------------------------------------ -- | Test whether a line is empty. isLineEmpty :: (Metric v, OrderedField n) => Trail' Line v n -> Bool isLineEmpty (Line (SegTree t)) = FT.null t -- | Test whether a trail is empty. Note that loops are never empty. isTrailEmpty :: (Metric v, OrderedField n) => Trail v n -> Bool isTrailEmpty = withTrail isLineEmpty (const False) -- | Determine whether a trail is a line. isLine :: Trail v n -> Bool isLine = not . isLoop -- | Determine whether a trail is a loop. isLoop :: Trail v n -> Bool isLoop = withTrail (const False) (const True) -- | Extract the segments comprising a line. lineSegments :: Trail' Line v n -> [Segment Closed v n] lineSegments (Line (SegTree t)) = F.toList t -- | Modify a line by applying a function to its list of segments. onLineSegments :: (Metric v, OrderedField n) => ([Segment Closed v n] -> [Segment Closed v n]) -> Trail' Line v n -> Trail' Line v n onLineSegments f = lineFromSegments . f . lineSegments -- | Extract the segments comprising a loop: a list of closed -- segments, and one final open segment. loopSegments :: Trail' Loop v n -> ([Segment Closed v n], Segment Open v n) loopSegments (Loop (SegTree t) c) = (F.toList t, c) -- | Extract the segments of a trail. If the trail is a loop it will -- first have 'cutLoop' applied. trailSegments :: (Metric v, OrderedField n) => Trail v n -> [Segment Closed v n] trailSegments = withLine lineSegments -- | Extract the offsets of the segments of a trail. trailOffsets :: (Metric v, OrderedField n) => Trail v n -> [v n] trailOffsets = withLine lineOffsets -- | Compute the offset from the start of a trail to the end. Satisfies -- -- @ -- trailOffset === sumV . trailOffsets -- @ -- -- but is more efficient. -- -- <<diagrams/src_Diagrams_Trail_trailOffsetEx.svg#diagram=trailOffsetEx&width=300>> -- -- > trailOffsetEx = (strokeLine almostClosed <> showOffset) # centerXY # pad 1.1 -- > where showOffset = fromOffsets [trailOffset (wrapLine almostClosed)] -- > # strokeP # lc red trailOffset :: (Metric v, OrderedField n) => Trail v n -> v n trailOffset = withLine lineOffset -- | Extract the offsets of the segments of a line. lineOffsets :: Trail' Line v n -> [v n] lineOffsets = map segOffset . lineSegments -- | Extract the offsets of the segments of a loop. loopOffsets :: (Metric v, OrderedField n) => Trail' Loop v n -> [v n] loopOffsets = lineOffsets . cutLoop -- | Compute the offset from the start of a line to the end. (Note, -- there is no corresponding @loopOffset@ function because by -- definition it would be constantly zero.) lineOffset :: (Metric v, OrderedField n) => Trail' Line v n -> v n lineOffset (Line t) = trailMeasure zero (op TotalOffset . view oeOffset) t -- | Extract the points of a concretely located trail, /i.e./ the points -- where one segment ends and the next begins. Note that for loops, -- the starting point will /not/ be repeated at the end. If you -- want this behavior, you can use 'cutTrail' to make the loop into -- a line first, which happens to repeat the same point at the start -- and end, /e.g./ with @trailPoints . mapLoc cutTrail@. -- -- Note that it does not make sense to ask for the points of a -- 'Trail' by itself; if you want the points of a trail -- with the first point at, say, the origin, you can use -- @trailPoints . (\`at\` origin)@. -- -- This function allows you "observe" the fact that trails are -- implemented as lists of segments, which may be problematic if we -- want to think of trails as parametric vector functions. This also -- means that the behavior of this function may not be stable under -- future changes to the implementation of trails. For an -- unproblematic version which only yields vertices at which there -- is a sharp corner, excluding points where the trail is -- differentiable, see 'trailVertices'. -- -- This function is not re-exported from "Diagrams.Prelude"; to use -- it, import "Diagrams.Trail". trailPoints :: (Metric v, OrderedField n) => Located (Trail v n) -> [Point v n] trailPoints (viewLoc -> (p,t)) = withTrail (linePoints . (`at` p)) (loopPoints . (`at` p)) t -- | Extract the segment join points of a concretely located line. See -- 'trailPoints' for more information. -- -- This function allows you "observe" the fact that lines are -- implemented as lists of segments, which may be problematic if we -- want to think of lines as parametric vector functions. This also -- means that the behavior of this function may not be stable under -- future changes to the implementation of trails. For an -- unproblematic version which only yields vertices at which there -- is a sharp corner, excluding points where the trail is -- differentiable, see 'lineVertices'. -- -- This function is not re-exported from "Diagrams.Prelude"; to use -- it, import "Diagrams.Trail". linePoints :: (Metric v, OrderedField n) => Located (Trail' Line v n) -> [Point v n] linePoints (viewLoc -> (p,t)) = segmentPoints p . lineSegments $ t -- | Extract the segment join points of a concretely located loop. Note that the -- initial vertex is not repeated at the end. See 'trailPoints' for -- more information. -- -- This function allows you "observe" the fact that lines are -- implemented as lists of segments, which may be problematic if we -- want to think of lines as parametric vector functions. This also -- means that the behavior of this function may not be stable under -- future changes to the implementation of trails. For an -- unproblematic version which only yields vertices at which there -- is a sharp corner, excluding points where the trail is -- differentiable, see 'lineVertices'. -- -- This function is not re-exported from "Diagrams.Prelude"; to use -- it, import "Diagrams.Trail". loopPoints :: (Metric v, OrderedField n) => Located (Trail' Loop v n) -> [Point v n] loopPoints (viewLoc -> (p,t)) = segmentPoints p . fst . loopSegments $ t segmentPoints :: (Additive v, Num n) => Point v n -> [Segment Closed v n] -> [Point v n] segmentPoints p = scanl (.+^) p . map segOffset tolerance :: OrderedField a => a tolerance = 10e-16 -- | Extract the vertices of a concretely located trail. Here a /vertex/ -- is defined as a non-differentiable point on the trail, /i.e./ a -- sharp corner. (Vertices are thus a subset of the places where -- segments join; if you want all joins between segments, see -- 'trailPoints'.) The tolerance determines how close the tangents -- of two segments must be at their endpoints to consider the -- transition point to be differentiable. -- -- Note that for loops, the starting vertex will /not/ be repeated -- at the end. If you want this behavior, you can use 'cutTrail' to -- make the loop into a line first, which happens to repeat the same -- vertex at the start and end, /e.g./ with @trailVertices . mapLoc -- cutTrail@. -- -- It does not make sense to ask for the vertices of a 'Trail' by -- itself; if you want the vertices of a trail with the first vertex -- at, say, the origin, you can use @trailVertices . (\`at\` -- origin)@. trailVertices' :: (Metric v, OrderedField n) => n -> Located (Trail v n) -> [Point v n] trailVertices' toler (viewLoc -> (p,t)) = withTrail (lineVertices' toler . (`at` p)) (loopVertices' toler . (`at` p)) t -- | Like 'trailVertices'', with a default tolerance. trailVertices :: (Metric v, OrderedField n) => Located (Trail v n) -> [Point v n] trailVertices = trailVertices' tolerance -- | Extract the vertices of a concretely located line. See -- 'trailVertices' for more information. lineVertices' :: (Metric v, OrderedField n) => n -> Located (Trail' Line v n) -> [Point v n] lineVertices' toler (viewLoc -> (p,t)) = segmentVertices' toler p . lineSegments $ t -- | Like 'lineVertices'', with a default tolerance. lineVertices :: (Metric v, OrderedField n) => Located (Trail' Line v n) -> [Point v n] lineVertices = lineVertices' tolerance -- | Extract the vertices of a concretely located loop. Note that the -- initial vertex is not repeated at the end. See 'trailVertices' for -- more information. loopVertices' :: (Metric v, OrderedField n) => n -> Located (Trail' Loop v n) -> [Point v n] loopVertices' toler (viewLoc -> (p,t)) | length segs > 1 = if far > toler then init ps else init . drop 1 $ ps | otherwise = ps where far = quadrance ((signorm . tangentAtStart . head $ segs) ^-^ (signorm . tangentAtEnd . last $ segs)) segs = lineSegments . cutLoop $ t ps = segmentVertices' toler p segs -- | Same as 'loopVertices'', with a default tolerance. loopVertices :: (Metric v, OrderedField n) => Located (Trail' Loop v n) -> [Point v n] loopVertices = loopVertices' tolerance -- | The vertices of a list of segments laid end to end. -- The start and end points are always included in the list of -- vertices. The other points connecting segments are included if -- the slope at the end of a segment is not equal to the slope at -- the beginning of the next. The 'toler' parameter is used to -- control how close the slopes need to be in order to declare them -- equal. segmentVertices' :: (Metric v, OrderedField n) => n -> Point v n -> [Segment Closed v n] -> [Point v n] segmentVertices' toler p ts = case ps of (x:_:_) -> x : select (drop 1 ps) ds ++ [last ps] _ -> ps where ds = zipWith far tans (drop 1 tans) tans = [(signorm . tangentAtStart $ s ,signorm . tangentAtEnd $ s) | s <- ts] ps = scanl (.+^) p . map segOffset $ ts far p2 q2 = quadrance (snd p2 ^-^ fst q2) > toler select :: [a] -> [Bool] -> [a] select xs bs = map fst $ filter snd (zip xs bs) -- | Convert a concretely located trail into a list of fixed segments. fixTrail :: (Metric v, OrderedField n) => Located (Trail v n) -> [FixedSegment v n] fixTrail t = map mkFixedSeg (trailLocSegments t) -- | Convert a concretely located trail into a list of located segments. trailLocSegments :: (Metric v, OrderedField n) => Located (Trail v n) -> [Located (Segment Closed v n)] trailLocSegments t = zipWith at (trailSegments (unLoc t)) (trailPoints t) ------------------------------------------------------------ -- Modifying trails -------------------------------------- ------------------------------------------------------------ -- | Reverse a trail. Semantically, if a trail given by a function t -- from [0,1] to vectors, then the reverse of t is given by t'(s) = -- t(1-s). @reverseTrail@ is an involution, that is, -- -- @ -- reverseTrail . reverseTrail === id -- @ reverseTrail :: (Metric v, OrderedField n) => Trail v n -> Trail v n reverseTrail = onTrail reverseLine reverseLoop -- | Reverse a concretely located trail. The endpoint of the original -- trail becomes the starting point of the reversed trail, so the -- original and reversed trails comprise exactly the same set of -- points. @reverseLocTrail@ is an involution, /i.e./ -- -- @ -- reverseLocTrail . reverseLocTrail === id -- @ reverseLocTrail :: (Metric v, OrderedField n) => Located (Trail v n) -> Located (Trail v n) reverseLocTrail (viewLoc -> (p, t)) = reverseTrail t `at` (p .+^ trailOffset t) -- | Reverse a line. See 'reverseTrail'. reverseLine :: (Metric v, OrderedField n) => Trail' Line v n -> Trail' Line v n reverseLine = onLineSegments (reverse . map reverseSegment) -- | Reverse a concretely located line. See 'reverseLocTrail'. reverseLocLine :: (Metric v, OrderedField n) => Located (Trail' Line v n) -> Located (Trail' Line v n) reverseLocLine (viewLoc -> (p,l)) = reverseLine l `at` (p .+^ lineOffset l) -- | Reverse a loop. See 'reverseTrail'. reverseLoop :: (Metric v, OrderedField n) => Trail' Loop v n -> Trail' Loop v n reverseLoop = glueLine . reverseLine . cutLoop -- | Reverse a concretely located loop. See 'reverseLocTrail'. Note -- that this is guaranteed to preserve the location. reverseLocLoop :: (Metric v, OrderedField n) => Located (Trail' Loop v n) -> Located (Trail' Loop v n) reverseLocLoop = mapLoc reverseLoop -- | Same as 'reverseLine' or 'reverseLoop'. instance (Metric v, OrderedField n) => Reversing (Trail' l v n) where reversing t@(Line _) = onLineSegments (reverse . map reversing) t reversing t@(Loop _ _) = glueLine . reversing . cutLoop $ t -- | Same as 'reverseTrail'. instance (Metric v, OrderedField n) => Reversing (Trail v n) where reversing (Trail t) = Trail (reversing t) -- | Same as 'reverseLocLine' or 'reverseLocLoop'. instance (Metric v, OrderedField n) => Reversing (Located (Trail' l v n)) where reversing l@(Loc _ Line {}) = reverseLocLine l reversing l@(Loc _ Loop {}) = reverseLocLoop l -- | Same as 'reverseLocTrail'. instance (Metric v, OrderedField n) => Reversing (Located (Trail v n)) where reversing = reverseLocTrail