{-# LANGUAGE DeriveFunctor #-} {-# LANGUAGE EmptyDataDecls #-} {-# LANGUAGE FlexibleContexts #-} {-# LANGUAGE FlexibleInstances #-} {-# LANGUAGE GADTs #-} {-# LANGUAGE GeneralizedNewtypeDeriving #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE StandaloneDeriving #-} {-# LANGUAGE TemplateHaskell #-} {-# LANGUAGE TypeFamilies #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE UndecidableInstances #-} {-# LANGUAGE ViewPatterns #-} {-# OPTIONS_GHC -fno-warn-orphans #-} -- We have an orphan Transformable FingerTree instance here. ----------------------------------------------------------------------------- -- | -- Module : Diagrams.Trail -- Copyright : (c) 2013 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(..) , wrapTrail, wrapLine, wrapLoop , onTrail, onLine , glueTrail, closeTrail, cutTrail -- * Constructing trails , emptyLine, emptyTrail , lineFromVertices, trailFromVertices , lineFromOffsets, trailFromOffsets , lineFromSegments, trailFromSegments -- * Eliminating trails , withTrail', withTrail, withLine , isLineEmpty, isTrailEmpty , isLine, isLoop , trailSegments, lineSegments, loopSegments , onLineSegments , trailOffsets, trailOffset , lineOffsets, lineOffset, loopOffsets , 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 (AnIso', iso, view, op, Wrapped(..), Rewrapped) import Data.AffineSpace import Data.FingerTree (FingerTree, ViewL (..), ViewR (..), (<|), (|>)) import qualified Data.FingerTree as FT import qualified Data.Foldable as F import Data.Monoid.MList import Data.Semigroup import Data.VectorSpace hiding (Sum (..)) import qualified Numeric.Interval.Kaucher as I import Diagrams.Core hiding ((|>)) import Diagrams.Located import Diagrams.Parametric import Diagrams.Segment -- $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 instance ( HasLinearMap (V a), InnerSpace (V a), OrderedField (Scalar (V a)) , FT.Measured m a, Transformable a ) => Transformable (FingerTree m a) where transform = FT.fmap' . transform ------------------------------------------------------------ -- 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 = SegTree (FingerTree (SegMeasure v) (Segment Closed v)) deriving (Eq, Ord, Show) instance Wrapped (SegTree v) where type Unwrapped (SegTree v) = FingerTree (SegMeasure v) (Segment Closed v) _Wrapped' = iso (\(SegTree x) -> x) SegTree instance Rewrapped (SegTree v) (SegTree v') type instance V (SegTree v) = v deriving instance (OrderedField (Scalar v), InnerSpace v) => Monoid (SegTree v) deriving instance (OrderedField (Scalar v), InnerSpace v) => FT.Measured (SegMeasure v) (SegTree v) deriving instance (HasLinearMap v, InnerSpace v, OrderedField (Scalar v)) => Transformable (SegTree v) type instance Codomain (SegTree v) = v instance (InnerSpace v, OrderedField (Scalar v), RealFrac (Scalar v)) => Parametric (SegTree v) where atParam t p = offset . fst $ splitAtParam t p instance Num (Scalar v) => DomainBounds (SegTree v) instance (InnerSpace v, OrderedField (Scalar v), RealFrac (Scalar v), Num (Scalar v)) => EndValues (SegTree v) instance (InnerSpace v, RealFrac (Scalar v), Floating (Scalar v)) => Sectionable (SegTree v) where splitAtParam (SegTree t) p | p < 0 = case FT.viewl t of EmptyL -> emptySplit seg :< t' -> case seg `splitAtParam` (p * tSegs) of (seg1, seg2) -> ( SegTree $ FT.singleton seg1 , SegTree $ seg2 <| t' ) | p >= 1 = case FT.viewr t of EmptyR -> emptySplit t' :> seg -> case seg `splitAtParam` (1 - (1 - p)*tSegs) of (seg1, seg2) -> ( SegTree $ t' |> seg1 , SegTree $ FT.singleton seg2 ) | otherwise = case FT.viewl after of EmptyL -> emptySplit seg :< after' -> case seg `splitAtParam` (snd . propFrac $ p * tSegs) of (seg1, seg2) -> ( SegTree $ before |> seg1 , SegTree $ seg2 <| after' ) where (before, after) = FT.split ((p * tSegs <) . numSegs) t tSegs = numSegs t emptySplit = (SegTree t, SegTree t) reverseDomain (SegTree t) = SegTree $ FT.reverse t' where t' = FT.fmap' reverseSegment t -- XXX seems like it should be possible to collapse some of the -- above cases into one? instance (InnerSpace v, OrderedField (Scalar v), RealFrac (Scalar v)) => HasArcLength (SegTree v) 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 :: ArcLength v -> I.Interval (Scalar v)) t fun = trailMeasure (const 0) (getArcLengthFun :: ArcLength v -> Scalar v -> I.Interval (Scalar v)) t arcLengthToParam eps st@(SegTree t) l | l < 0 = case FT.viewl t of EmptyL -> 0 seg :< _ -> arcLengthToParam eps seg l / tSegs | l >= totalAL = case FT.viewr t of EmptyR -> 0 t' :> 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 :< _ -> 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) (Segment Closed v) (before, after) = FT.split ((>= l) . trailMeasure 0 (I.midpoint . (getArcLengthBounded eps :: ArcLength v -> I.Interval (Scalar v)))) 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 :: ( InnerSpace v, OrderedField (Scalar v) , SegMeasure v :>: m, FT.Measured (SegMeasure v) 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 :: ( Floating (Scalar v), Num c, Ord (Scalar v), InnerSpace v, FT.Measured (SegMeasure v) a ) => a -> c numSegs = fromIntegral . trailMeasure 0 (getSum . op SegCount) -- | Compute the total offset of anything measured by 'SegMeasure'. offset :: ( Floating (Scalar v), Ord (Scalar v), InnerSpace v, FT.Measured (SegMeasure v) t ) => t -> v offset = trailMeasure zeroV (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 where Line :: SegTree v -> Trail' Line v Loop :: SegTree v -> Segment Open v -> Trail' Loop v -- | A generic eliminator for 'Trail'', taking functions specifying -- what to do in the case of a line or a loop. withTrail' :: (Trail' Line v -> r) -> (Trail' Loop v -> r) -> Trail' l v -> r withTrail' line _ t@(Line{}) = line t withTrail' _ loop t@(Loop{}) = loop t deriving instance Show v => Show (Trail' l v) deriving instance Eq v => Eq (Trail' l v) deriving instance Ord v => Ord (Trail' l v) type instance V (Trail' l v) = v type instance Codomain (Trail' l v) = v instance (OrderedField (Scalar v), InnerSpace v) => Semigroup (Trail' Line v) 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 (OrderedField (Scalar v), InnerSpace v) => Monoid (Trail' Line v) where mempty = emptyLine mappend = (<>) instance (HasLinearMap v, InnerSpace v, OrderedField (Scalar v)) => Transformable (Trail' l v) 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 (InnerSpace v, OrderedField (Scalar v)) => Enveloped (Trail' l v) where getEnvelope = withTrail' ftEnv (ftEnv . cutLoop) where ftEnv :: Trail' Line v -> Envelope v ftEnv (Line t) = trailMeasure mempty (view oeEnvelope) $ t instance (HasLinearMap v, InnerSpace v, OrderedField (Scalar v)) => Renderable (Trail' o v) NullBackend where render _ _ = mempty instance (InnerSpace v, OrderedField (Scalar v), RealFrac (Scalar v)) => Parametric (Trail' l v) where atParam t p = withTrail' (\(Line segT) -> segT `atParam` p) (\l -> cutLoop l `atParam` mod1 p) t -- | Compute the remainder mod 1. Convenient for constructing loop -- parameterizations that wrap around. mod1 :: RealFrac a => a -> a mod1 p = p' where pf = snd . propFrac $ p p' | p >= 0 = pf | otherwise = 1 + pf -- Get rid of defaulting warnings propFrac :: RealFrac a => a -> (Int, a) propFrac = properFraction instance Num (Scalar v) => DomainBounds (Trail' l v) instance (InnerSpace v, OrderedField (Scalar v), RealFrac (Scalar v)) => EndValues (Trail' l v) instance (InnerSpace v, RealFrac (Scalar v), Floating (Scalar v)) => Sectionable (Trail' Line v) where splitAtParam (Line t) p = (Line t1, Line t2) where (t1, t2) = splitAtParam t p reverseDomain = reverseLine instance (InnerSpace v, OrderedField (Scalar v), RealFrac (Scalar v)) => HasArcLength (Trail' l v) 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 -------------------------------------------------- -- 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' (Scalar v) (Scalar v))@. @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 -- | 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 Codomain (GetSegment t) = Maybe ( V t -- offset from trail start to segment start , Segment Closed (V t) -- the segment , AnIso' (Scalar (V t)) (Scalar (V t)) -- reparameterization, trail <-> segment ) -- | 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 (InnerSpace v, OrderedField (Scalar v)) => Parametric (GetSegment (Trail' Line v)) where atParam (GetSegment (Line (SegTree ft))) p | p <= 0 = case FT.viewl ft of EmptyL -> Nothing seg :< _ -> Just (zeroV, seg, reparam 0) | p >= 1 = case FT.viewr ft of EmptyR -> Nothing ft' :> seg -> Just (offset ft', seg, reparam (n-1)) | otherwise = let (before, after) = FT.split ((p*n <) . numSegs) $ ft in case FT.viewl after of EmptyL -> Nothing seg :< _ -> 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 (InnerSpace v, OrderedField (Scalar v), RealFrac (Scalar v)) => Parametric (GetSegment (Trail' Loop v)) where atParam (GetSegment l) p = atParam (GetSegment (cutLoop l)) (mod1 p) instance (InnerSpace v, OrderedField (Scalar v), RealFrac (Scalar v)) => Parametric (GetSegment (Trail v)) 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 (InnerSpace v, OrderedField (Scalar v)) => EndValues (GetSegment (Trail' Line v)) where atStart (GetSegment (Line (SegTree ft))) = case FT.viewl ft of EmptyL -> Nothing seg :< _ -> let n = numSegs ft in Just (zeroV, seg, iso (*n) (/n)) atEnd (GetSegment (Line (SegTree ft))) = case FT.viewr ft of EmptyR -> Nothing ft' :> seg -> let n = numSegs ft in Just (offset ft', seg, iso (subtract (n-1) . (*n)) ((/n) . (+ (n-1))) ) instance (InnerSpace v, OrderedField (Scalar v), RealFrac (Scalar v)) => EndValues (GetSegment (Trail' Loop v)) where atStart (GetSegment l) = atStart (GetSegment (cutLoop l)) atEnd (GetSegment l) = atEnd (GetSegment (cutLoop l)) instance (InnerSpace v, OrderedField (Scalar v), RealFrac (Scalar v)) => EndValues (GetSegment (Trail v)) where atStart (GetSegment t) = withTrail (\l -> atStart (GetSegment l)) (\l -> atStart (GetSegment l)) t atEnd (GetSegment t) = withTrail (\l -> atEnd (GetSegment l)) (\l -> atEnd (GetSegment l)) 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 where Trail :: Trail' l v -> Trail v deriving instance Show v => Show (Trail v) instance Eq v => Eq (Trail v) where t1 == t2 = withTrail (\ln1 -> withTrail (\ln2 -> ln1 == ln2) (const False) t2) (\lp1 -> withTrail (const False) (\lp2 -> lp1 == lp2) t2) t1 instance Ord v => Ord (Trail v) where compare t1 t2 = withTrail (\ln1 -> withTrail (\ln2 -> compare ln1 ln2) (const LT) t2) (\lp1 -> withTrail (const GT) (\lp2 -> compare lp1 lp2) 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 (Scalar v), InnerSpace v) => Semigroup (Trail v) 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 (OrderedField (Scalar v), InnerSpace v) => Monoid (Trail v) where mempty = wrapLine emptyLine mappend = (<>) type instance V (Trail v) = v type instance Codomain (Trail v) = v instance (HasLinearMap v, InnerSpace v, OrderedField (Scalar v)) => Transformable (Trail v) where transform t = onTrail (transform t) (transform t) instance (InnerSpace v, OrderedField (Scalar v)) => Enveloped (Trail v) where getEnvelope = withTrail getEnvelope getEnvelope instance (InnerSpace v, OrderedField (Scalar v), RealFrac (Scalar v)) => Parametric (Trail v) where atParam t p = withTrail (`atParam` p) (`atParam` p) t instance Num (Scalar v) => DomainBounds (Trail v) instance (InnerSpace v, OrderedField (Scalar v), RealFrac (Scalar v)) => EndValues (Trail v) -- | 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 (InnerSpace v, RealFrac (Scalar v), Floating (Scalar v)) => Sectionable (Trail v) where splitAtParam t p = withLine ((wrapLine *** wrapLine) . (`splitAtParam` p)) t reverseDomain = reverseTrail instance (InnerSpace v, OrderedField (Scalar v), RealFrac (Scalar v)) => HasArcLength (Trail v) where arcLengthBounded = withLine . arcLengthBounded arcLengthToParam eps tr al = withLine (\ln -> arcLengthToParam eps ln al) tr -------------------------------------------------- -- 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 -> r) -> (Trail' Loop v -> r) -> Trail v -> 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 -> Trail' l1 v) -> (Trail' Loop v -> Trail' l2 v) -> (Trail v -> Trail v) 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 :: (InnerSpace v, OrderedField (Scalar v)) => (Trail' Line v -> r) -> Trail v -> 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 :: (InnerSpace v, OrderedField (Scalar v)) => (Trail' Line v -> Trail' Line v) -> Trail v -> Trail v 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 -> Trail v 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 -> Trail v 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 -> Trail v wrapLoop = wrapTrail ------------------------------------------------------------ -- Constructing trails ----------------------------------- ------------------------------------------------------------ -- | The empty line, which is the identity for concatenation of lines. emptyLine :: (InnerSpace v, OrderedField (Scalar v)) => Trail' Line v emptyLine = Line mempty -- | A wrapped variant of 'emptyLine'. emptyTrail :: (InnerSpace v, OrderedField (Scalar v)) => Trail v emptyTrail = wrapLine emptyLine -- | Construct a line from a list of closed segments. lineFromSegments :: (InnerSpace v, OrderedField (Scalar v)) => [Segment Closed v] -> Trail' Line v lineFromSegments = Line . SegTree . FT.fromList -- | @trailFromSegments === 'wrapTrail' . 'lineFromSegments'@, for -- conveniently constructing a @Trail@ instead of a @Trail'@. trailFromSegments :: (InnerSpace v, OrderedField (Scalar v)) => [Segment Closed v] -> Trail v 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 :: (InnerSpace v, OrderedField (Scalar v)) => [v] -> Trail' Line v lineFromOffsets = lineFromSegments . map straight -- | @trailFromOffsets === 'wrapTrail' . 'lineFromOffsets'@, for -- conveniently constructing a @Trail@ instead of a @Trail' Line@. trailFromOffsets :: (InnerSpace v, OrderedField (Scalar v)) => [v] -> Trail v 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 :: (InnerSpace v, OrderedField (Scalar v)) => [Point v] -> Trail' Line v 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 :: (InnerSpace v, OrderedField (Scalar v)) => [Point v] -> Trail v 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 . hcat' (with & sep .~ 1) -- > $ [almostClosed # strokeLine, almostClosed # glueLine # strokeLoop] -- > -- > almostClosed :: Trail' Line R2 -- > 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 :: (InnerSpace v, OrderedField (Scalar v)) => Trail' Line v -> Trail' Loop v glueLine (Line (SegTree t)) = case FT.viewr t of FT.EmptyR -> Loop mempty (Linear OffsetOpen) t' :> (Linear _) -> Loop (SegTree t') (Linear OffsetOpen) t' :> (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 :: (InnerSpace v, OrderedField (Scalar v)) => Trail v -> Trail v 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 -> Trail' Loop v 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 -> Trail v 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. (InnerSpace v, OrderedField (Scalar v)) => Trail' Loop v -> Trail' Line v 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 offV = negateV . trailMeasure zeroV (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 :: (InnerSpace v, OrderedField (Scalar v)) => Trail v -> Trail v cutTrail = onTrail id cutLoop ------------------------------------------------------------ -- Eliminating trails ------------------------------------ ------------------------------------------------------------ -- | Test whether a line is empty. isLineEmpty :: (InnerSpace v, OrderedField (Scalar v)) => Trail' Line v -> Bool isLineEmpty (Line (SegTree t)) = FT.null t -- | Test whether a trail is empty. Note that loops are never empty. isTrailEmpty :: (InnerSpace v, OrderedField (Scalar v)) => Trail v -> Bool isTrailEmpty = withTrail isLineEmpty (const False) -- | Determine whether a trail is a line. isLine :: Trail v -> Bool isLine = not . isLoop -- | Determine whether a trail is a loop. isLoop :: Trail v -> Bool isLoop = withTrail (const False) (const True) -- | Extract the segments comprising a line. lineSegments :: Trail' Line v -> [Segment Closed v] lineSegments (Line (SegTree t)) = F.toList t -- | Modify a line by applying a function to its list of segments. onLineSegments :: (InnerSpace v, OrderedField (Scalar v)) => ([Segment Closed v] -> [Segment Closed v]) -> Trail' Line v -> Trail' Line v 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 -> ([Segment Closed v], Segment Open v) 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 :: (InnerSpace v, OrderedField (Scalar v)) => Trail v -> [Segment Closed v] trailSegments = withLine lineSegments -- | Extract the offsets of the segments of a trail. trailOffsets :: (InnerSpace v, OrderedField (Scalar v)) => Trail v -> [v] 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)] -- > # stroke # lc red trailOffset :: (InnerSpace v, OrderedField (Scalar v)) => Trail v -> v trailOffset = withLine lineOffset -- | Extract the offsets of the segments of a line. lineOffsets :: (InnerSpace v, OrderedField (Scalar v)) => Trail' Line v -> [v] lineOffsets = map segOffset . lineSegments -- | Extract the offsets of the segments of a loop. loopOffsets :: (InnerSpace v, OrderedField (Scalar v)) => Trail' Loop v -> [v] 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 :: (InnerSpace v, OrderedField (Scalar v)) => Trail' Line v -> v lineOffset (Line t) = trailMeasure zeroV (op TotalOffset . view oeOffset) t -- | Extract the vertices of a concretely located trail. 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@. -- -- Note that 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 :: (InnerSpace v, OrderedField (Scalar v)) => Located (Trail v) -> [Point v] trailVertices (viewLoc -> (p,t)) = withTrail (lineVertices . (`at` p)) (loopVertices . (`at` p)) t -- | Extract the vertices of a concretely located line. See -- 'trailVertices' for more information. lineVertices :: (InnerSpace v, OrderedField (Scalar v)) => Located (Trail' Line v) -> [Point v] lineVertices (viewLoc -> (p,t)) = segmentVertices p . lineSegments $ t -- | 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 :: (InnerSpace v, OrderedField (Scalar v)) => Located (Trail' Loop v) -> [Point v] loopVertices (viewLoc -> (p,t)) = segmentVertices p . fst . loopSegments $ t segmentVertices :: AdditiveGroup v => Point v -> [Segment Closed v] -> [Point v] segmentVertices p = scanl (.+^) p . map segOffset -- | Convert a concretely located trail into a list of fixed segments. fixTrail :: (InnerSpace v, OrderedField (Scalar v)) => Located (Trail v) -> [FixedSegment v] fixTrail t = map mkFixedSeg (trailLocSegments t) -- | Convert a concretely located trail into a list of located segments. trailLocSegments :: (InnerSpace v, OrderedField (Scalar v)) => Located (Trail v) -> [Located (Segment Closed v)] trailLocSegments t = zipWith at (trailSegments (unLoc t)) (trailVertices 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 :: (InnerSpace v, OrderedField (Scalar v)) => Trail v -> Trail v 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 :: (InnerSpace v, OrderedField (Scalar v)) => Located (Trail v) -> Located (Trail v) reverseLocTrail (viewLoc -> (p, t)) = reverseTrail t `at` (p .+^ trailOffset t) -- | Reverse a line. See 'reverseTrail'. reverseLine :: (InnerSpace v, OrderedField (Scalar v)) => Trail' Line v -> Trail' Line v reverseLine = onLineSegments (reverse . map reverseSegment) -- | Reverse a concretely located line. See 'reverseLocTrail'. reverseLocLine :: (InnerSpace v, OrderedField (Scalar v)) => Located (Trail' Line v) -> Located (Trail' Line v) reverseLocLine (viewLoc -> (p,l)) = reverseLine l `at` (p .+^ lineOffset l) -- | Reverse a loop. See 'reverseTrail'. reverseLoop :: (InnerSpace v, OrderedField (Scalar v)) => Trail' Loop v -> Trail' Loop v reverseLoop = glueLine . reverseLine . cutLoop -- | Reverse a concretely located loop. See 'reverseLocTrail'. Note -- that this is guaranteed to preserve the location. reverseLocLoop :: (InnerSpace v, OrderedField (Scalar v)) => Located (Trail' Loop v) -> Located (Trail' Loop v) reverseLocLoop = mapLoc reverseLoop