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compute brush strokes using the envelope equation

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sheaf 2023-01-09 20:54:19 +01:00
parent 43098dec01
commit 8fc5e6c9b8
17 changed files with 683 additions and 460 deletions
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4
MetaBrush.cabal
View file

@ -36,6 +36,8 @@ common common
>= 4.17 && < 4.18
, acts
^>= 0.3.1.0
, code-page
^>= 0.2.1
, containers
>= 0.6.0.1 && < 0.7
, deepseq
@ -112,6 +114,7 @@ common common
common extras
build-depends:
directory
>= 1.3.4.0 && < 1.4
, filepath
@ -180,6 +183,7 @@ library splines
, Math.Module
, Math.Orientation
, Math.Roots
, Debug.Utils
build-depends:
bifunctors

8
app/Main.hs
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@ -1,6 +1,4 @@
{-# LANGUAGE BlockArguments #-}
{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE TypeApplications #-}
module Main
( main )
@ -14,6 +12,10 @@ import System.Exit
import GHC.Conc
( getNumProcessors, setNumCapabilities )
-- code-page
import System.IO.CodePage
( withCP65001 )
-- gi-gio
import qualified GI.Gio as GIO
@ -30,7 +32,7 @@ import MetaBrush.Application
--------------------------------------------------------------------------------
main :: IO ()
main = do
main = withCP65001 do
procs <- getNumProcessors
let

2
src/app/MetaBrush/Action.hs
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@ -821,7 +821,7 @@ instance HandleAction MouseClick where
changeText :: Text
changeText = "Subdivide " <> loc
pure ( UpdateDoc . UpdateDocumentTo $ HistoryChange {..} )
-- Ignore double click event otherwise.
_ -> pure Don'tModifyDoc

19
src/app/MetaBrush/Application.hs
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@ -84,7 +84,6 @@ import Math.Linear
import MetaBrush.Action
( ActionOrigin(..) )
import qualified MetaBrush.Asset.Brushes as Asset.Brushes
( EllipseBrushFields, ellipse )
import MetaBrush.Asset.Colours
( getColours )
import MetaBrush.Asset.Logo
@ -161,6 +160,12 @@ runApplication application = do
, strokeUnique = strokeUnique
, strokeBrush = Just Asset.Brushes.ellipse
, strokeSpline =
-- Spline
-- { splineStart = mkPoint ( ℝ2 -20 -20 ) 5
-- , splineCurves = OpenCurves $ Seq.fromList
-- [ LineTo { curveEnd = NextPoint ( mkPoint ( ℝ2 20 20 ) 5 ), curveData = invalidateCache undefined }
-- ]
-- }
Spline
{ splineStart = mkPoint ( ℝ2 10 -20 ) 2 1 0
, splineCurves = OpenCurves $ Seq.fromList
@ -176,6 +181,8 @@ runApplication application = do
where
mkPoint :: ℝ 2 -> Double -> Double -> Double -> PointData ( Record Asset.Brushes.EllipseBrushFields )
mkPoint pt a b phi = PointData pt Normal ( MkR $ ℝ3 a b phi )
-- mkPoint :: ℝ 2 -> Double -> PointData ( Record Asset.Brushes.CircleBrushFields )
-- mkPoint pt r = PointData pt Normal ( MkR $ ℝ1 r )
recomputeStrokesTVar <- STM.newTVarIO @Bool False
documentRenderTVar <- STM.newTVarIO @( ( Int32, Int32 ) -> Cairo.Render () ) ( const $ pure () )
@ -193,11 +200,11 @@ runApplication application = do
maxHistorySizeTVar <- STM.newTVarIO @Int 1000
fitParametersTVar <- STM.newTVarIO @FitParameters
( FitParameters
{ maxSubdiv = 6
, nbSegments = 12
, dist_tol = 5e-3
, t_tol = 1e-4
, maxIters = 100
{ maxSubdiv = 2 --3 -- 6
, nbSegments = 3 --6 -- 12
, dist_tol = 0.1 -- 5e-3
, t_tol = 0.1 -- 1e-4
, maxIters = 2 -- 100
}
)

13
src/app/MetaBrush/Render/Document.hs
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@ -292,17 +292,18 @@ strokeRenderData fitParams
{ inject
, project = toUsedParams :: Record pointFields -> Record usedFields }
-> do
let embedUsedParams r = inject r brush_defaults
let embedUsedParams = inject brush_defaults
-- Compute the outline using the brush function.
( outline, fitPts ) <-
computeStrokeOutline @( T ( Record usedFields) ) @clo
fitParams ( toUsedParams . brushParams ) ( fun brushFn . embedUsedParams ) spline
computeStrokeOutline @clo fitParams
( toUsedParams . brushParams ) embedUsedParams brushFn
spline
pure $
StrokeWithOutlineRenderData
{ strokeDataSpline = spline
, strokeOutlineData = ( outline, fitPts )
, strokeBrushFunction = fun brushFn . embedUsedParams . toUsedParams
, strokeBrushFunction = fun brushFn . fun embedUsedParams . toUsedParams
}
_ -> pure $
StrokeRenderData
@ -604,7 +605,7 @@ drawFitPoint _ zoom ( FitPoint { fitPoint = ℝ2 x y } ) = do
lift do
Cairo.save
Cairo.translate x y
Cairo.arc 0 0 ( 2 / zoom ) 0 ( 2 * pi )
Cairo.arc 0 0 ( 6 / zoom ) 0 ( 2 * pi ) -- 2 / zoom
Cairo.setSourceRGBA r g b 0.8
Cairo.fill
Cairo.restore
@ -621,7 +622,7 @@ drawFitPoint _ zoom ( FitTangent { fitPoint = ℝ2 x y, fitTangent = V2 tx ty }
Cairo.translate x y
Cairo.moveTo 0 0
Cairo.lineTo ( 0.05 * tx ) ( 0.05 * ty )
Cairo.setLineWidth ( 1 / zoom )
Cairo.setLineWidth ( 4 / zoom ) -- 1 / zoom
Cairo.setSourceRGBA r g b 0.8
Cairo.stroke
Cairo.arc 0 0 ( 2 / zoom ) 0 ( 2 * pi )

4
src/app/MetaBrush/UI/ToolBar.hs
View file

@ -55,12 +55,12 @@ data Mode
data ToolBar
= ToolBar
{ selectionTool, penTool, pathTool, brushTool, metaTool, debugTool
:: !GTK.ToggleButton
:: !GTK.ToggleButton
}
createToolBar :: Variables -> Colours -> GTK.Box -> IO ToolBar
createToolBar ( Variables {..} ) colours toolBar = do
widgetAddClass toolBar "toolBar"
GTK.widgetSetValign toolBar GTK.AlignStart

41
src/metabrushes/MetaBrush/Asset/Brushes.hs
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@ -3,10 +3,6 @@
module MetaBrush.Asset.Brushes where
-- base
import Data.Coerce
( coerce )
-- containers
import qualified Data.Sequence as Seq
( fromList )
@ -24,9 +20,9 @@ import qualified Data.HashMap.Strict as HashMap
-- MetaBrush
import Math.Bezier.Spline
import Math.Linear
( ℝ(..), T(..) )
( ℝ(..), Fin(..) )
import Math.Linear.Dual
( D, type (~>)(..), Var(var), konst )
( D, type (~>)(..), var, konst )
import Math.Module
( Module((^+^), (*^)) )
import MetaBrush.Brush
@ -86,17 +82,16 @@ circleBrush :: Record CircleBrushFields ~> Spline 'Closed () ( ℝ 2 )
circleBrush =
D \ params ->
let r :: D ( Record CircleBrushFields ) Double
r = runD ( var @1 ) params
r = runD ( var ( Fin 1## ) ) params
mkPt :: Double -> Double -> D ( Record CircleBrushFields ) ( ℝ 2 )
mkPt ( kon -> x ) ( kon -> y )
= fmap coerce
$ ( x * r ) *^ e_x
= ( x * r ) *^ e_x
^+^ ( y * r ) *^ e_y
in circleSpline @( Record CircleBrushFields ) mkPt
where
e_x, e_y :: D ( Record CircleBrushFields ) ( T ( ℝ 2 ) )
e_x = pure $ T $ ℝ2 1 0
e_y = pure $ T $ ℝ2 0 1
e_x, e_y :: D ( Record CircleBrushFields ) ( ℝ 2 )
e_x = pure $ ℝ2 1 0
e_y = pure $ ℝ2 0 1
kon = konst @( Record CircleBrushFields )
@ -104,25 +99,17 @@ ellipseBrush :: Record EllipseBrushFields ~> Spline 'Closed () ( ℝ 2 )
ellipseBrush =
D \ params ->
let a, b, phi :: D ( Record EllipseBrushFields ) Double
a = runD ( var @1 ) params
b = runD ( var @2 ) params
phi = runD ( var @3 ) params
a = runD ( var ( Fin 1## ) ) params
b = runD ( var ( Fin 2## ) ) params
phi = runD ( var ( Fin 3## ) ) params
mkPt :: Double -> Double -> D ( Record EllipseBrushFields ) ( ℝ 2 )
mkPt ( kon -> x ) ( kon -> y )
= fmap coerce
$ ( x * a * cos phi - y * b * sin phi ) *^ e_x
= ( x * a * cos phi - y * b * sin phi ) *^ e_x
^+^ ( y * b * cos phi + x * a * sin phi ) *^ e_y
in circleSpline @( Record EllipseBrushFields ) mkPt
where
e_x, e_y :: D ( Record EllipseBrushFields ) ( T ( ℝ 2 ) )
e_x = pure $ T $ ℝ2 1 0
e_y = pure $ T $ ℝ2 0 1
e_x, e_y :: D ( Record EllipseBrushFields ) ( ℝ 2 )
e_x = pure $ ℝ2 1 0
e_y = pure $ ℝ2 0 1
kon = konst @( Record EllipseBrushFields )
--ellipseArc :: ℝ 2 ~> ℝ 2
--ellipseArc = brushStroke ( linear myPath ) ( uncurryD $ fmap bezier3 myBrush )
--testing :: Double -> Double -> (# Double, T ( ℝ 2 ) #)
--testing :: Double -> Double -> (# Double, T (ℝ 2) #)
--testing t s = envelopeEquation ellipseArc t s

12
src/metabrushes/MetaBrush/Brush.hs
View file

@ -1,6 +1,7 @@
{-# LANGUAGE AllowAmbiguousTypes #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE AllowAmbiguousTypes #-}
{-# LANGUAGE QuantifiedConstraints #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE UndecidableInstances #-}
module MetaBrush.Brush
( Brush(..), SomeBrush(..), BrushFunction
@ -38,12 +39,17 @@ import qualified Data.Text as Text
-- MetaBrush
import Math.Linear
( ℝ, Representable )
import Math.Linear.Dual
( Diffy )
import Math.Module
( Interpolatable )
import Math.Bezier.Spline
( SplineType(Closed), SplinePts)
import MetaBrush.Records
( KnownSymbols, Length, Record, WithParams )
import MetaBrush.Serialisable
( Serialisable )
--------------------------------------------------------------------------------

4
src/metabrushes/MetaBrush/Document/SubdivideStroke.hs
View file

@ -49,7 +49,7 @@ import Math.Bezier.Spline
import Math.Bezier.Stroke
( CachedStroke(..), invalidateCache )
import Math.Module
( lerp, quadrance, closestPointOnSegment )
( Interpolatable, lerp, quadrance, closestPointOnSegment )
import Math.Linear
( Segment(..), ℝ(..), T(..) )
import MetaBrush.Document
@ -57,8 +57,6 @@ import MetaBrush.Document
, PointData(..), DiffPointData(..)
, coords, _strokeSpline
)
import MetaBrush.Records
( Interpolatable )
--------------------------------------------------------------------------------

35
src/metabrushes/MetaBrush/Records.hs
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@ -54,13 +54,6 @@ import Math.Module
--------------------------------------------------------------------------------
-- | A convenient constraint synonym for types that support interpolation.
type Interpolatable :: Type -> Constraint
class ( Torsor ( T r ) r, Module Double ( T r ) ) => Interpolatable r
instance ( Torsor ( T r ) r, Module Double ( T r ) ) => Interpolatable r
--------------------------------------------------------------------------------
-- | A function from a given record type, with provided default values
-- that can be overridden.
type WithParams :: [ Symbol ] -> Type -> Type
@ -123,9 +116,11 @@ instance ( Torsor ( T ( ℝ ( Length ks ) ) ) ( ℝ ( Length ks ) )
=> Torsor ( T ( Record ks ) ) ( Record ks ) where
MkR g --> MkR a = T $ MkR $ unT $ g --> a
type instance D ( Record ks ) = D ( ℝ ( Length ks ) )
deriving newtype
instance Representable ( ℝ ( Length ks ) )
=> Representable ( Record ks )
deriving newtype instance Var n ( ℝ ( Length ks ) ) => Var n ( Record ks )
type instance D ( Record ks ) = D ( ℝ ( Length ks ) )
deriving newtype instance Diffy ( ℝ ( Length ks ) ) => Diffy ( Record ks )
--------------------------------------------------------------------------------
@ -154,31 +149,35 @@ intersect :: forall r1 r2 l1 l2
, KnownSymbols r1, KnownSymbols r2
, l1 ~ Length r1, l2 ~ Length r2
, Representable ( ℝ l1 ), Representable ( ℝ l2 )
, Interpolatable ( Record r1 )
, Interpolatable ( Record r1 ), Diffy ( ℝ l2 )
)
=> Intersection r1 r2
intersect
-- Shortcut when the two rows are equal.
| Just Refl <- eqT @r1 @r2
, Refl <- ( unsafeCoerce Refl :: r1 :~: Intersect r1 r2 )
= Intersection { project = id, inject = const }
= Intersection { project = id, inject = \ _ -> linear id }
| otherwise
= doIntersection @r1 @r2 \ ( _ :: Proxy# r1r2 ) r1_idxs r2_idxs ->
let
project :: Record r1 -> Record r1r2
project = \ ( MkR r1 ) -> MkR $ projection ( (!) r1_idxs ) r1
inject :: Record r1r2 -> Record r2 -> Record r2
inject = \ ( MkR r1r2 ) ( MkR r2 ) -> MkR $ injection ( find eqFin r2_idxs ) r1r2 r2
inject :: Record r2 -> Record r1r2 ~> Record r2
inject = \ ( MkR r2 ) -> linear \ ( MkR r1r2 ) -> MkR $ injection ( find eqFin r2_idxs ) r1r2 r2
in Intersection { project, inject }
data Intersection r1 r2 where
Intersection
:: forall r1r2 r1 r2
. ( KnownSymbols r1r2, Representable ( ℝ ( Length r1r2 ) )
:: forall r1r2 r1 r2 l12
. ( l12 ~ Length r1r2
, KnownSymbols r1r2
, Representable ( ℝ l12 )
, Diffy ( ℝ l12 )
, Interpolatable ( Record r1r2 ) )
=> { project :: Record r1 -> Record r1r2
, inject :: Record r1r2 -> Record r2 -> Record r2
=> { project :: Record r1 -> Record r1r2
, inject :: Record r2 -> Record r1r2 ~> Record r2
-- ^ overrides the components of the first record with the second
} -> Intersection r1 r2
{-# INLINE doIntersection #-}
@ -190,7 +189,7 @@ doIntersection
)
=> ( forall r1r2 l12.
( r1r2 ~ Intersect r1 r2, l12 ~ Length r1r2
, Representable ( ℝ l12 ), Interpolatable ( ℝ l12 )
, Representable ( ℝ l12 ), Diffy ( ℝ l12 ), Interpolatable ( ℝ l12 )
, KnownSymbols r1r2, Representable ( ℝ ( Length r1r2 ) )
)
=> Proxy# r1r2 -> Vec l12 ( Fin l1 ) -> Vec l12 ( Fin l2 ) -> kont )

26
src/splines/Debug/Utils.hs Normal file
View file

@ -0,0 +1,26 @@
module Debug.Utils ( trace ) where
-- base
import System.IO
( BufferMode(..), hSetBuffering, hFlush, hPutStrLn, stdout )
import System.IO.Unsafe
( unsafePerformIO )
-- deepseq
import Control.DeepSeq
( force )
-- code-page
import System.IO.CodePage
( withCP65001 )
--------------------------------------------------------------------------------
trace :: String -> a -> a
trace ( force -> !str ) expr =
unsafePerformIO $ withCP65001 do
hSetBuffering stdout ( BlockBuffering $ Just 50 )
hPutStrLn stdout str
hFlush stdout
return expr

14
src/splines/Math/Bezier/Cubic/Fit.hs
View file

@ -120,7 +120,7 @@ data FitPoint
-- including the meaning of \( \texttt{t_tol} \) and \( \texttt{maxIters} \).
fitSpline
:: FitParameters
-> ( Double -> ( ℝ 2, T ( ℝ 2 ) ) ) -- ^ curve \( t \mapsto ( C(t), C'(t) ) \) to fit
-> ( ℝ 1 -> ( ℝ 2, T ( ℝ 2 ) ) ) -- ^ curve \( t \mapsto ( C(t), C'(t) ) \) to fit
-> ( SplinePts Open, Seq FitPoint )
fitSpline ( FitParameters {..} ) = go 0
where
@ -128,16 +128,16 @@ fitSpline ( FitParameters {..} ) = go 0
dt = recip ( fromIntegral nbSegments )
go
:: Int
-> ( Double -> ( ℝ 2, T ( ℝ 2 ) ) )
-> ( ℝ 1 -> ( ℝ 2, T ( ℝ 2 ) ) )
-> ( SplinePts Open, Seq FitPoint )
go subdiv curveFn =
let
p, r :: ℝ 2
tp, tr :: T ( ℝ 2 )
qs :: [ ℝ 2 ]
(p, tp) = curveFn 0
(r, tr) = curveFn 1
qs = [ fst $ curveFn ( dt * fromIntegral j ) | j <- [ 1 .. nbSegments - 1 ] ]
(p, tp) = curveFn $ ℝ1 0
(r, tr) = curveFn $ ℝ1 1
qs = [ fst $ curveFn ( ℝ1 $ dt * fromIntegral j ) | j <- [ 1 .. nbSegments - 1 ] ]
in
case fitPiece dist_tol t_tol maxIters p tp qs r tr of
( bez, Max ( Arg max_sq_error t_split ) )
@ -150,8 +150,8 @@ fitSpline ( FitParameters {..} ) = go 0
c1, c2 :: SplinePts Open
ps1, ps2 :: Seq FitPoint
( ( c1, ps1 ), ( c2, ps2 ) )
= ( go ( subdiv + 1 ) ( \ t -> curveFn $ t * t_split_eff )
, go ( subdiv + 1 ) ( \ t -> curveFn $ t_split_eff + t * ( 1 - t_split_eff ) )
= ( go ( subdiv + 1 ) ( \ ( ℝ1 t ) -> curveFn $ ℝ1 $ t * t_split_eff )
, go ( subdiv + 1 ) ( \ ( ℝ1 t ) -> curveFn $ ℝ1 $ t_split_eff + t * ( 1 - t_split_eff ) )
) `Parallel.Strategy.using`
( Parallel.Strategy.parTuple2 Parallel.Strategy.rdeepseq Parallel.Strategy.rdeepseq )
-> ( c1 <> c2, ps1 <> ps2 )

531
src/splines/Math/Bezier/Stroke.hs
View file

@ -1,5 +1,6 @@
{-# LANGUAGE AllowAmbiguousTypes #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE AllowAmbiguousTypes #-}
{-# LANGUAGE QuantifiedConstraints #-}
{-# LANGUAGE ScopedTypeVariables #-}
module Math.Bezier.Stroke
( Offset(..)
@ -9,10 +10,8 @@ module Math.Bezier.Stroke
-- * Brush stroking
-- $brushes
, brushStroke, envelopeEquation
, linear, bezier2, bezier3
-- , uncurryD
, line, bezier2, bezier3
)
where
@ -22,6 +21,8 @@ import Prelude
hiding ( unzip )
import Control.Arrow
( first, (***) )
import Control.Applicative
( Applicative(..) )
import Control.Monad
( unless )
import Control.Monad.ST
@ -30,6 +31,8 @@ import Data.Bifunctor
( Bifunctor(bimap) )
import Data.Foldable
( for_ )
import Data.Functor.Identity
( Identity(..) )
import Data.List.NonEmpty
( unzip )
import Data.Maybe
@ -53,11 +56,15 @@ import Data.Act
import Data.Sequence
( Seq(..) )
import qualified Data.Sequence as Seq
--( empty, reverse, singleton, index )
import Data.Set
( Set )
import qualified Data.Set as Set
( singleton )
-- deepseq
import Control.DeepSeq
( NFData(..), NFData1, deepseq )
( NFData(..), NFData1, deepseq, force )
-- generic-lens
import Data.Generics.Product.Typed
@ -65,10 +72,6 @@ import Data.Generics.Product.Typed
import Data.Generics.Internal.VL
( set, view )
-- groups
import Data.Group
( Group )
-- parallel
import qualified Control.Parallel.Strategies as Strats
( rdeepseq, parTuple2, using )
@ -101,8 +104,8 @@ import qualified Math.Bezier.Quadratic as Quadratic
import Math.Epsilon
( epsilon )
import Math.Module
( Module(..), Inner((^.^))
, lerp, squaredNorm, cross
( Module(..), Inner((^.^)), Interpolatable
, lerp, cross
, convexCombination, strictlyParallel
)
import Math.Orientation
@ -110,10 +113,12 @@ import Math.Orientation
, between
)
import Math.Roots
( solveQuadratic )
( solveQuadratic, newtonRaphson )
import Math.Linear
import Math.Linear.Dual
--import Debug.Utils
--------------------------------------------------------------------------------
data Offset
@ -168,42 +173,42 @@ coords = view typed
--------------------------------------------------------------------------------
type OutlineFn = ℝ 1 -> ( ( ℝ 2, T ( ℝ 2 ) ), ( ℝ 2, T ( ℝ 2 ) ) )
computeStrokeOutline ::
forall diffParams ( clo :: SplineType ) brushParams crvData ptData s
forall ( clo :: SplineType ) usedParams brushParams crvData ptData s
. ( KnownSplineType clo
, Group diffParams
, Module Double diffParams
, Torsor diffParams brushParams
, Interpolatable usedParams
, Diffy usedParams, Diffy brushParams
, HasType ( ℝ 2 ) ptData
, HasType ( CachedStroke s ) crvData
, NFData ptData, NFData crvData
-- Debugging.
, Show ptData
, Show ptData, Show brushParams
)
=> FitParameters
-> ( ptData -> brushParams )
-> ( brushParams -> SplinePts Closed )
-> ( ptData -> usedParams )
-> ( usedParams ~> brushParams )
-> ( brushParams ~> SplinePts Closed )
-> Spline clo crvData ptData
-> ST s
( Either ( SplinePts Closed ) ( SplinePts Closed, SplinePts Closed )
, Seq FitPoint
)
computeStrokeOutline fitParams ptParams brushFn spline@( Spline { splineStart = spt0 } ) = case ssplineType @clo of
computeStrokeOutline fitParams ptParams toBrushParams brushFn spline@( Spline { splineStart = spt0 } ) = case ssplineType @clo of
-- Open brush path with at least one segment.
-- Need to add caps at both ends of the path.
SOpen
| firstCurve :<| _ <- openCurves $ splineCurves spline
, prevCurves :|> lastCurve <- openCurves $ splineCurves spline
, ( firstOutlineFwd, firstOutlineBwd ) :<| _ <- outlineFns
, _ :|> ( lastOutlineFwd, lastOutlineBwd ) <- outlineFns
, firstOutlineFn :<| _ <- outlineFns
, _ :|> lastOutlineFn <- outlineFns
, let
endPt :: ptData
endPt = openCurveEnd lastCurve
startTgtFwd, startTgtBwd, endTgtFwd, endTgtBwd :: T ( ℝ 2 )
startTgtFwd = snd ( firstOutlineFwd 0 )
startTgtBwd = -1 *^ snd ( firstOutlineBwd 1 )
endTgtFwd = snd ( lastOutlineFwd 1 )
endTgtBwd = -1 *^ snd ( lastOutlineBwd 0 )
( ( _, startTgtFwd), ( _, startTgtBwd ) ) = firstOutlineFn $ ℝ1 0
( ( _, endTgtFwd ), ( _, endTgtBwd ) ) = lastOutlineFn $ ℝ1 1
startBrush, endBrush :: SplinePts Closed
startBrush = brushShape spt0
endBrush = brushShape endPt
@ -252,8 +257,8 @@ computeStrokeOutline fitParams ptParams brushFn spline@( Spline { splineStart =
-- Add forward and backward caps at the start.
SClosed
| ClosedCurves prevCurves lastCurve <- splineCurves spline
, ( firstOutlineFwd, firstOutlineBwd ) :<| _ <- outlineFns
, _ :|> ( lastOutlineFwd, lastOutlineBwd ) <- outlineFns
, firstOutlineFn :<| _ <- outlineFns
, _ :|> lastOutlineFn <- outlineFns
, let
startTgt, endTgt, startTgtFwd, startTgtBwd, endTgtFwd, endTgtBwd :: T ( ℝ 2 )
startTgt = case prevCurves of
@ -262,10 +267,8 @@ computeStrokeOutline fitParams ptParams brushFn spline@( Spline { splineStart =
endTgt = case prevCurves of
Empty -> endTangent spt0 spt0 lastCurve
_ :|> lastPrev -> endTangent spt0 ( openCurveEnd lastPrev ) lastCurve
startTgtFwd = snd ( firstOutlineFwd 0 )
startTgtBwd = -1 *^ snd ( firstOutlineBwd 1 )
endTgtFwd = snd ( lastOutlineFwd 1 )
endTgtBwd = -1 *^ snd ( lastOutlineBwd 0 )
( ( _, startTgtFwd), ( _, startTgtBwd ) ) = firstOutlineFn $ ℝ1 0
( ( _, endTgtFwd ), ( _, endTgtBwd ) ) = lastOutlineFn $ ℝ1 1
fwdStartCap, bwdStartCap :: SplinePts Open
TwoSided fwdStartCap bwdStartCap
= fmap fst . snd . runWriter
@ -284,26 +287,19 @@ computeStrokeOutline fitParams ptParams brushFn spline@( Spline { splineStart =
)
where
outlineFns
:: Seq
( Double -> ( ℝ 2, T ( ℝ 2 ) )
, Double -> ( ℝ 2, T ( ℝ 2 ) )
)
outlineFns :: Seq OutlineFn
outlineFns = go spt0 ( openCurves $ splineCurves ( adjustSplineType @Open spline ) )
where
go
:: ptData
-> Seq ( Curve Open crvData ptData )
-> Seq
( Double -> ( ℝ 2, T ( ℝ 2 ) )
, Double -> ( ℝ 2, T ( ℝ 2 ) )
)
-> Seq OutlineFn
go _ Empty = Empty
go p0 ( crv :<| crvs ) =
outlineFunctions @diffParams ptParams brushFn p0 crv :<| go ( openCurveEnd crv ) crvs
outlineFunction ptParams toBrushParams brushFn p0 crv :<| go ( openCurveEnd crv ) crvs
brushShape :: ptData -> SplinePts Closed
brushShape pt = brushFn ( ptParams pt )
brushShape pt = fun brushFn $ fun toBrushParams $ ptParams pt
updateSpline :: ( T ( ℝ 2 ), T ( ℝ 2 ), T ( ℝ 2 ) ) -> ST s OutlineData
updateSpline ( lastTgt, lastTgtFwd, lastTgtBwd )
@ -313,17 +309,15 @@ computeStrokeOutline fitParams ptParams brushFn spline@( Spline { splineStart =
( \ ptData curve -> do
( prevTgt, prev_tgtFwd, prev_tgtBwd ) <- get
let
fwd, bwd :: Double -> ( ℝ 2, T ( ℝ 2 ) )
( fwd, bwd ) = outlineFunctions @diffParams ptParams brushFn ptData curve
fwdBwd :: OutlineFn
fwdBwd = outlineFunction ptParams toBrushParams brushFn ptData curve
tgt, next_tgt, tgtFwd, next_tgtFwd, tgtBwd, next_tgtBwd :: T ( ℝ 2 )
tgt = startTangent spt0 ptData curve
next_tgt = endTangent spt0 ptData curve
tgtFwd = snd ( fwd 0 )
next_tgtFwd = snd ( fwd 1 )
tgtBwd = -1 *^ snd ( bwd 1 )
next_tgtBwd = -1 *^ snd ( bwd 0 )
tgt = startTangent spt0 ptData curve
next_tgt = endTangent spt0 ptData curve
( ( _, tgtFwd ), ( _, tgtBwd ) ) = fwdBwd $ ℝ1 0
( ( _, next_tgtFwd ), ( _, next_tgtBwd ) ) = fwdBwd $ ℝ1 1
lift $ tellBrushJoin ( prevTgt, prev_tgtFwd, tgtBwd ) ptData ( tgt, tgtFwd, prev_tgtBwd )
lift $ updateCurveData ( curveData curve ) fwd bwd
lift $ updateCurveData ( curveData curve ) fwdBwd
put ( next_tgt, next_tgtFwd, next_tgtBwd )
)
( const ( pure () ) )
@ -331,10 +325,9 @@ computeStrokeOutline fitParams ptParams brushFn spline@( Spline { splineStart =
updateCurveData
:: crvData
-> ( Double -> ( ℝ 2, T ( ℝ 2 ) ) )
-> ( Double -> ( ℝ 2, T ( ℝ 2 ) ) )
-> OutlineFn
-> WriterT OutlineData ( ST s ) ()
updateCurveData ( view ( typed @( CachedStroke s ) ) -> CachedStroke { cachedStrokeRef } ) fwd bwd = do
updateCurveData ( view ( typed @( CachedStroke s ) ) -> CachedStroke { cachedStrokeRef } ) fwdBwd = do
mbOutline <- lift ( readSTRef cachedStrokeRef )
case mbOutline of
-- Cached fit data is available: use it.
@ -346,12 +339,12 @@ computeStrokeOutline fitParams ptParams brushFn spline@( Spline { splineStart =
let
fwdData, bwdData :: ( SplinePts Open, Seq FitPoint )
( fwdData, bwdData ) =
( fitSpline fitParams fwd, fitSpline fitParams bwd )
( fitSpline fitParams ( fst . fwdBwd ), fitSpline fitParams ( snd . fwdBwd ) )
`Strats.using`
( Strats.parTuple2 Strats.rdeepseq Strats.rdeepseq )
outlineData :: OutlineData
outlineData = TwoSided fwdData bwdData
outlineData `deepseq` tell ( outlineData )
outlineData = TwoSided fwdData ( bimap reverseSpline Seq.reverse bwdData )
outlineData `deepseq` tell outlineData
lift $ writeSTRef cachedStrokeRef ( Just outlineData )
-- Connecting paths at a point of discontinuity of the tangent vector direction (G1 discontinuity).
@ -416,107 +409,55 @@ computeStrokeOutline fitParams ptParams brushFn spline@( Spline { splineStart =
$ joinWithBrush brush0 tgtBwd prevTgtBwd
-- | Computes the forward and backward stroke outline functions for a single curve.
outlineFunctions
:: forall diffParams brushParams crvData ptData
. ( Group diffParams, Module Double diffParams
, Torsor diffParams brushParams
outlineFunction
:: forall usedParams brushParams crvData ptData
. ( Interpolatable usedParams
, Diffy usedParams, Diffy brushParams
, HasType ( ℝ 2 ) ptData
-- Debugging.
, Show ptData
, Show ptData, Show brushParams
)
=> ( ptData -> brushParams )
-> ( brushParams -> SplinePts Closed )
=> ( ptData -> usedParams )
-> ( usedParams ~> brushParams )
-> ( brushParams ~> SplinePts Closed )
-> ptData
-> Curve Open crvData ptData
-> ( Double -> ( ℝ 2, T ( ℝ 2 ) )
, Double -> ( ℝ 2, T ( ℝ 2 ) )
)
outlineFunctions ptParams brushFn sp0 crv =
-> OutlineFn
outlineFunction ptParams toBrushParams brushFromParams sp0 crv =
let
p0 :: ℝ 2
p0 = coords sp0
brush :: Double -> SplinePts Closed
f :: Double -> ℝ 2
f' :: Double -> T ( ℝ 2 )
( brush, f, f' ) = case crv of
LineTo { curveEnd = NextPoint sp1 }
| let
p1 :: ℝ 2
p1 = coords sp1
tgt :: T ( ℝ 2 )
tgt = p0 --> p1
brush1 :: Double -> SplinePts Closed
brush1 t = brushFn ( lerp @diffParams t ( ptParams sp0 ) ( ptParams sp1 ) )
-> ( brush1, \ t -> lerp @( T ( ℝ 2 ) ) t p0 p1, const tgt )
Bezier2To { controlPoint = sp1, curveEnd = NextPoint sp2 }
| let
p1, p2 :: ℝ 2
p1 = coords sp1
p2 = coords sp2
bez :: Quadratic.Bezier ( ℝ 2 )
bez = Quadratic.Bezier {..}
brush2 :: Double -> SplinePts Closed
brush2 t =
brushFn $
Quadratic.bezier @diffParams
( Quadratic.Bezier ( ptParams sp0 ) ( ptParams sp1 ) ( ptParams sp2 ) ) t
-> ( brush2, Quadratic.bezier @( T ( ℝ 2 ) ) bez, Quadratic.bezier' bez )
Bezier3To { controlPoint1 = sp1, controlPoint2 = sp2, curveEnd = NextPoint sp3 }
| let
p1, p2, p3 :: ℝ 2
p1 = coords sp1
p2 = coords sp2
p3 = coords sp3
bez :: Cubic.Bezier ( ℝ 2 )
bez = Cubic.Bezier {..}
brush3 :: Double -> SplinePts Closed
brush3 t =
brushFn $
Cubic.bezier @diffParams
( Cubic.Bezier ( ptParams sp0 ) ( ptParams sp1 ) ( ptParams sp2 ) ( ptParams sp3 ) ) t
-> ( brush3, Cubic.bezier @( T ( ℝ 2 ) ) bez, Cubic.bezier' bez )
fwd, bwd :: Double -> ( ℝ 2, T ( ℝ 2 ) )
fwd t
= ( off t --offset ( withTangent ( fwd' t ) ( brush t ) ) • f t
, fwd' t
)
usedParams :: ℝ 1 ~> usedParams
path :: ℝ 1 ~> ℝ 2
( path, usedParams ) =
case crv of
LineTo { curveEnd = NextPoint sp1 }
| let seg = Segment sp0 sp1
-> ( line ( fmap coords seg ), line ( fmap ptParams seg ) )
Bezier2To { controlPoint = sp1, curveEnd = NextPoint sp2 }
| let bez2 = Quadratic.Bezier sp0 sp1 sp2
-> ( bezier2 ( fmap coords bez2 ), bezier2 ( fmap ptParams bez2 ) )
Bezier3To { controlPoint1 = sp1, controlPoint2 = sp2, curveEnd = NextPoint sp3 }
| let bez3 = Cubic.Bezier sp0 sp1 sp2 sp3
-> ( bezier3 ( fmap coords bez3 ), bezier3 ( fmap ptParams bez3 ) )
fwdBwd :: OutlineFn
fwdBwd t
= solveEnvelopeEquations path_t path'_t ( fwdOffset, bwdOffset ) curves
-- = ( ( offset fwdOffset • path_t, path'_t )
-- , ( offset bwdOffset • path_t, -1 *^ path'_t ) )
where
off :: Double -> ℝ 2
off u = offset ( withTangent ( f' u ) ( brush u ) ) • f u
offTgt :: Double -> T ( ℝ 2 )
offTgt u
| u < 0.5
= 1e9 *^ ( off u --> off (u + 1e-9) )
| otherwise
= 1e9 *^ ( off (u - 1e-9) --> off u )
fwd' :: Double -> T ( ℝ 2 )
fwd' u
| squaredNorm ( offTgt u ) < epsilon
= f' u
| otherwise
= offTgt u
bwd t
= ( off s --offset ( withTangent ( -1 *^ bwd' s ) ( brush s ) ) • f s
, bwd' s
)
where
s :: Double
s = 1 - t
off :: Double -> ℝ 2
off u = offset ( withTangent ( -1 *^ f' u ) ( brush u ) ) • f u
offTgt :: Double -> T ( ℝ 2 )
offTgt u
| u < 0.5
= 1e9 *^ ( off u --> off (u + 1e-9) )
| otherwise
= 1e9 *^ ( off (u - 1e-9) --> off u )
bwd' :: Double -> T ( ℝ 2 )
bwd' u
| squaredNorm ( offTgt u ) < epsilon
= -1 *^ f' u
| otherwise
= offTgt u
in ( fwd, bwd )
curves :: Seq ( ℝ 1 -> StrokeDatum )
curves = brushStrokeData path ( usedParams `chainRule` toBrushParams ) brushFromParams t
fwdOffset = withTangent path'_t brush_t
bwdOffset = withTangent ( -1 *^ path'_t ) brush_t
D1 path_t path'_t _ = runD path t
D1 params_t _ _ = runD usedParams t
brush_t = value @brushParams
$ runD brushFromParams ( fun toBrushParams params_t )
in fwdBwd
-----------------------------------
-- Various utility functions
@ -616,7 +557,7 @@ discardCurveData = bimap ( const () ) coords
dropFirstPiece :: HasType ( ℝ 2 ) ptData => Spline Open crvData ptData -> Maybe ( SplinePts Open )
dropFirstPiece ( Spline { splineCurves = OpenCurves curves } ) = case curves of
Empty -> Nothing
fstPiece :<| laterPieces ->
fstPiece :<| laterPieces ->
Just $ Spline
{ splineStart = coords ( openCurveEnd fstPiece )
, splineCurves = OpenCurves $ fmap discardCurveData laterPieces
@ -799,38 +740,6 @@ withTangent tgt_wanted spline@( Spline { splineStart } )
--------------------------------------------------------------------------------
-- $brushes
--
-- You can compute the envelope equation for a brush stroke by using
-- the functions 'linear', 'bezier2' and 'bezier3' in conjunction with
-- the 'brushStroke' function, e.g.
--
-- > brushStroke ( bezier2 path ) ( uncurryD $ fmap bezier3 brush )
-- | Linear interpolation, as a differentiable function.
linear :: forall b. ( Module Double ( T b ), Torsor ( T b ) b )
=> Segment b -> ℝ 1 ~> b
linear ( Segment a b ) = D \ ( ℝ1 t ) ->
D1 ( lerp @( T b ) t a b )
( a --> b )
origin
-- | A quadratic Bézier curve, as a differentiable function.
bezier2 :: forall b. ( Module Double ( T b ), Torsor ( T b ) b )
=> Quadratic.Bezier b -> ℝ 1 ~> b
bezier2 bez = D \ ( ℝ1 t ) ->
D1 ( Quadratic.bezier @( T b ) bez t )
( Quadratic.bezier' bez t )
( Quadratic.bezier'' bez )
-- | A cubic Bézier curve, as a differentiable function.
bezier3 :: forall b. ( Module Double ( T b ), Torsor ( T b ) b )
=> Cubic.Bezier b -> ℝ 1 ~> b
bezier3 bez = D \ ( ℝ1 t ) ->
D1 ( Cubic.bezier @( T b ) bez t )
( Cubic.bezier' bez t )
( Cubic.bezier'' bez t )
-- | A brush stroke, as described by the equation
--
-- \[ c(t,s) = p(t) + b(t,s) \]
@ -839,15 +748,10 @@ bezier3 bez = D \ ( ℝ1 t ) ->
--
-- - \( p(t) \) is the path that the brush follows, and
-- - \( b(t,s) \) is the brush shape, as it varies along the path.
brushStroke :: ℝ 1 ~> ℝ 2 -- ^ stroke path \( p(t) \)
-> ℝ 2 ~> ℝ 2 -- ^ brush \( b(t,s) \)
-> ℝ 2 ~> ℝ 2
brushStroke ( D f_p ) ( D f_b ) = D \ ( ℝ2 t0 s0 ) ->
let !( D1 p dpdt d2pdt2 )
= f_p ( ℝ1 t0 )
!( D2 b dbdt dbds d2bdt2 d2bdtds d2bds2 )
= f_b ( ℝ2 t0 s0 )
in
brushStroke :: D ( ℝ 1 ) ( ℝ 2 ) -- ^ stroke path \( p(t) \)
-> D ( ℝ 2 ) ( ℝ 2 ) -- ^ brush \( b(t,s) \)
-> D ( ℝ 2 ) ( ℝ 2 )
brushStroke ( D1 p dpdt d2pdt2 ) ( D2 b dbdt dbds d2bdt2 d2bdtds d2bds2 ) =
D2 ( unT $ T p ^+^ T b )
-- c = p + b
@ -868,14 +772,16 @@ brushStroke ( D f_p ) ( D f_b ) = D \ ( ℝ2 t0 s0 ) ->
--
-- \[ \frac{\partial E}{\partial s} \frac{\mathrm{d} c}{\mathrm{d} t} \]
--
-- whose roots correspond to cusps in the envelope.
envelopeEquation :: ℝ 2 ~> ℝ 2 -> Double -> Double -> (# Double, T ( ℝ 2 ) #)
envelopeEquation ( D c ) t s =
case c ( ℝ2 t s ) of
D2 _ dcdt dcds d2cdt2 d2cdtds d2cds2 ->
let dEdt = d2cdt2 `cross2` dcds + dcdt `cross2` d2cdtds
dEds = d2cdtds `cross2` dcds + dcdt `cross2` d2cds2
in (# dcdt `cross2` dcds, dEds *^ dcdt ^-^ dEdt *^ dcds #)
-- whose roots correspond to cusps in the envelope, and
--
-- \[ \frac{\partial E}{\partial s}. \]
envelopeEquation :: D ( ℝ 2 ) ( ℝ 2 ) -> ( Double, T ( ℝ 2 ), Double, Double )
envelopeEquation ( D2 _ dcdt dcds d2cdt2 d2cdtds d2cds2 ) =
let ee = dcdt `cross` dcds
dEdt = d2cdt2 `cross` dcds + dcdt `cross` d2cdtds
dEds = d2cdtds `cross` dcds + dcdt `cross` d2cds2
tot = dcdt ^-^ ( dEdt / dEds ) *^ dcds --dEds *^ dcdt ^-^ dEdt *^ dcds
in ( ee, tot, dEdt, dEds )
-- Computation of total derivative dc/dt:
--
-- dc/dt = ∂c/∂t + ∂c/∂s ∂s/∂t
@ -883,7 +789,218 @@ envelopeEquation ( D c ) t s =
--
-- ( ∂E / ∂s ) dc/dt = ( ∂E / ∂s ) ∂c/∂t - ( ∂E / ∂t ) ∂c/∂s.
-- | Cross-product of two 2D vectors.
cross2 :: T ( ℝ 2 ) -> T ( ℝ 2 ) -> Double
cross2 ( T ( ℝ2 x1 y1 ) ) ( T ( ℝ2 x2 y2 ) )
= x1 * y2 - x2 * y1
-- | Linear interpolation, as a differentiable function.
line :: forall b. ( Module Double ( T b ), Torsor ( T b ) b )
=> Segment b -> ℝ 1 ~> b
line ( Segment a b ) = D \ ( ℝ1 t ) ->
D1 ( lerp @( T b ) t a b )
( a --> b )
origin
-- | A quadratic Bézier curve, as a differentiable function.
bezier2 :: forall b. ( Module Double ( T b ), Torsor ( T b ) b )
=> Quadratic.Bezier b -> ℝ 1 ~> b
bezier2 bez = D \ ( ℝ1 t ) ->
D1 ( Quadratic.bezier @( T b ) bez t )
( Quadratic.bezier' bez t )
( Quadratic.bezier'' bez )
-- | A cubic Bézier curve, as a differentiable function.
bezier3 :: forall b. ( Module Double ( T b ), Torsor ( T b ) b )
=> Cubic.Bezier b -> ℝ 1 ~> b
bezier3 bez = D \ ( ℝ1 t ) ->
D1 ( Cubic.bezier @( T b ) bez t )
( Cubic.bezier' bez t )
( Cubic.bezier'' bez t )
splineCurveFns :: SplinePts Closed -> Seq ( ℝ 1 ~> ℝ 2 )
splineCurveFns spls
= runIdentity
. bifoldSpline
( \ pt crv -> Identity . Seq.singleton $ curveFn pt crv )
( const $ Identity Seq.empty )
. adjustSplineType @Open
$ spls
where
curveFn :: ℝ 2 -> Curve Open () ( ℝ 2 ) -> ( ℝ 1 ~> ℝ 2 )
curveFn p0 = \case
LineTo { curveEnd = NextPoint p1 }
-> line $ Segment p0 p1
Bezier2To { controlPoint = p1, curveEnd = NextPoint p2 }
-> bezier2 $ Quadratic.Bezier p0 p1 p2
Bezier3To { controlPoint1 = p1, controlPoint2 = p2, curveEnd = NextPoint p3 }
-> bezier3 $ Cubic.Bezier p0 p1 p2 p3
-- | Solve the envelope equations at a given point \( t = t_0 \), to find
-- \( s_0 \) such that \( c(t_0, s_0) \) is on the envelope of the brush stroke.
solveEnvelopeEquations :: ℝ 2
-> T ( ℝ 2 )
-> ( Offset, Offset )
-> Seq ( ℝ 1 -> StrokeDatum )
-> ( ( ℝ 2, T ( ℝ 2 ) ), ( ℝ 2, T ( ℝ 2 ) ) )
solveEnvelopeEquations path_t path'_t ( fwdOffset, bwdOffset ) strokeData
= ( fwdSol, ( bwdPt, -1 *^ bwdTgt ) )
where
-- !_ = trace
-- ( unlines
-- [ "solveEnvelopeEquation"
-- , " pt: " ++ show path_t
-- , " tgt: " ++ show path'_t
-- , " fwdOffset: " ++ show fwdOffset
-- , " bwdOffset: " ++ show bwdOffset ]
-- ) True
fwdSol = findSolFrom False fwdOffset
( bwdPt, bwdTgt ) = findSolFrom True bwdOffset
n :: Int
n = length strokeData
findSolFrom :: Bool -> Offset -> ( ℝ 2, T ( ℝ 2 ) )
findSolFrom goingBwds ( Offset { offsetIndex = i00, offsetParameter = s00, offset = off } )
= go ( Set.singleton i00 ) i00 ( fromMaybe 0.5 s00 )
where
plausibleTangent :: T ( ℝ 2 ) -> Bool
plausibleTangent tgt
| goingBwds
= path'_t ^.^ tgt < 0
| otherwise
= path'_t ^.^ tgt > 0
go :: Set Int -> Int -> Double -> ( ℝ 2, T ( ℝ 2 ) )
go seen i0 s0 =
case sol s0 ( strokeData `Seq.index` i0 ) of
( good, s, pt, tgt )
| True --good && plausibleTangent tgt
-> ( pt, tgt )
-- | let ( i', s0' ) = mbNextPoint i0 ( unℝ1 s )
-- , not ( i' `Set.member` seen )
-- -> go ( Set.insert i' seen ) i' s0'
| otherwise
-> ( off • path_t, path'_t )
mbNextPoint :: Int -> Double -> ( Int, Double )
mbNextPoint i0 s0
| s0 <= 0.5
= ( prev i0, 0.9 )
| otherwise
= ( next i0, 0.1 )
prev, next :: Int -> Int
prev i
| i == 0
= n - 1
| otherwise
= i - 1
next i
| i == n - 1
= 0
| otherwise
= i + 1
sol :: Double -> ( ℝ 1 -> StrokeDatum ) -> ( Bool, ℝ 1, ℝ 2, T ( ℝ 2 ) )
sol initialGuess f =
let (good, s) = case newtonRaphson maxIters precision domain ( eqn f ) initialGuess of
Nothing -> ( False, initialGuess )
Just s0 -> ( True , s0 )
in case f ( ℝ1 s ) of -- TODO: a bit redundant to have to compute this again...
StrokeDatum { dstroke, dcdt } ->
( good, ℝ1 s, value @( ℝ 2 ) dstroke, dcdt )
eqn :: ( ℝ 1 -> StrokeDatum ) -> ( Double -> ( Double, Double ) )
eqn f s =
case f ( ℝ1 s ) of
StrokeDatum { ee, 𝛿E𝛿s } ->
( ee, 𝛿E𝛿s )
maxIters :: Word
maxIters = 5 --30
precision :: Int
precision = 10
domain :: ( Double, Double )
domain = ( 0, 1 )
newtype ZipSeq a = ZipSeq { getZipSeq :: Seq a }
deriving stock Functor
instance Applicative ZipSeq where
pure _ = error "no pure for ZipSeq"
liftA2 f ( ZipSeq xs ) ( ZipSeq ys ) = ZipSeq ( Seq.zipWith f xs ys )
brushStrokeData :: forall brushParams
. ( Diffy brushParams, Show brushParams )
=> ( ℝ 1 ~> ℝ 2 ) -- ^ path
-> ( ℝ 1 ~> brushParams ) -- ^ brush parameters
-> ( brushParams ~> SplinePts Closed ) -- ^ brush from parameters
-> ( ℝ 1 -> Seq ( ℝ 1 -> StrokeDatum ) )
brushStrokeData path params brush =
\ t ->
let
dpath_t :: D ( ℝ 1 ) ( ℝ 2 )
!dpath_t = runD path t
dparams_t :: D ( ℝ 1 ) brushParams
!dparams_t@( D1 { v = params_t } ) = runD params t
dbrush_params :: D brushParams ( SplinePts Closed )
!dbrush_params = runD brush params_t
splines :: Seq ( D brushParams ( ℝ 1 ~> ℝ 2 ) )
!splines = getZipSeq $ traverse @_ @ZipSeq ( ZipSeq . splineCurveFns ) dbrush_params
dbrushes_t :: Seq ( ℝ 1 -> D ( ℝ 2 ) ( ℝ 2 ) )
!dbrushes_t = force $ fmap ( uncurryD' . ( dparams_t `chain` ) ) splines
in fmap ( mkStrokeDatum dpath_t ) dbrushes_t
where
mkStrokeDatum :: D ( ℝ 1 ) ( ℝ 2 )
-> ( ℝ 1 -> D ( ℝ 2 ) ( ℝ 2 ) )
-> ( ℝ 1 -> StrokeDatum )
mkStrokeDatum dpath_t dbrush_t s =
let dbrush_t_s = dbrush_t s
dstroke@( D2 _c _𝛿c𝛿t _𝛿c𝛿s _ _ _ ) = brushStroke dpath_t dbrush_t_s
( ee, dcdt, _𝛿E𝛿t, 𝛿E𝛿s ) = envelopeEquation dstroke
in -- trace
-- ( unlines
-- [ "envelopeEquation:"
-- , " s = " ++ show s
-- , " c = " ++ show _c
-- , " ∂c/∂t = " ++ show _𝛿c𝛿t
-- , " ∂c/∂s = " ++ show _𝛿c𝛿s
-- , " E = " ++ show ee
-- , " ∂E/∂t = " ++ show _𝛿E𝛿t
-- , " ∂E/∂s = " ++ show 𝛿E𝛿s
-- , " dc/dt = " ++ show dcdt ] ) $
StrokeDatum
{ dpath = dpath_t
, dbrush = dbrush_t_s
, dstroke
, ee, dcdt, 𝛿E𝛿s }
-- | The value and derivative of a brush stroke at a given coordinate
-- \( (t_0, s_0) \), together with the value of the envelope equation at that
-- point.
data StrokeDatum
= StrokeDatum
{ -- | Path \( p(t_0) \) (with its 1st and 2nd derivatives).
dpath :: D ( ℝ 1 ) ( ℝ 2 )
-- | Brush shape \( b(t_0, s_0) \) (with its 1st and 2nd derivatives).
, dbrush :: D ( ℝ 2 ) ( ℝ 2 )
-- Everything below can be computed in terms of the first two fields.
-- | Stroke \( c(t_0,s_0) = p(t_0) + b(t_0,s_0) \) (with its 1st and 2nd derivatives).
, dstroke :: D ( ℝ 2 ) ( ℝ 2 )
-- | Envelope
--
-- \[ E(t_0,s_0) = \left ( \frac{\partial c}{\partial t} \times \frac{\partial c}{\partial s} \right )_{(t_0,s_0)}. \]
, ee :: Double
-- | Total derivative
--
-- \[ \left ( \frac{\mathrm{d} c}{\mathrm{d} t} \right )_{(t_0,s_0)}. \]
, dcdt :: T ( ℝ 2 )
-- | \( \left ( \frac{\partial E}{\partial s} \right )_{(t_0,s_0)}. \)
, 𝛿E𝛿s :: Double
}
deriving stock Show

10
src/splines/Math/Linear.hs
View file

@ -71,7 +71,7 @@ data family ℝ n
data instance ℝ 0 = ℝ0
deriving stock ( Show, Eq, Ord, Generic )
deriving anyclass NFData
newtype instance ℝ 1 = ℝ1 Double
newtype instance ℝ 1 = ℝ1 { unℝ1 :: Double }
deriving stock ( Generic )
deriving newtype ( Show, Eq, Ord, NFData )
data instance ℝ 2 = ℝ2 {-# UNPACK #-} !Double {-# UNPACK #-} !Double
@ -86,8 +86,12 @@ data instance ℝ 3 = ℝ3 {-# UNPACK #-} !Double {-# UNPACK #-} !Double {-#
-- | Tangent space to Euclidean space.
type T :: Type -> Type
newtype T e = T { unT :: e }
deriving stock ( Eq, Functor )
deriving newtype ( Show, NFData ) -- newtype Show instance for debugging...
deriving stock ( Eq, Functor, Foldable, Traversable )
deriving newtype NFData -- newtype Show instance for debugging...
instance {-# OVERLAPPING #-} Show ( ℝ n ) => Show ( T ( ℝ n ) ) where
show ( T p ) = "V" ++ drop 1 ( show p )
deriving stock instance Show v => Show ( T v )
instance Applicative T where
pure = T

162
src/splines/Math/Linear/Dual.hs
View file

@ -1,5 +1,6 @@
{-# LANGUAGE AllowAmbiguousTypes #-}
{-# LANGUAGE DuplicateRecordFields #-}
{-# LANGUAGE QuantifiedConstraints #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE UndecidableInstances #-}
@ -8,12 +9,12 @@ module Math.Linear.Dual where
-- base
import Control.Applicative
( liftA2 )
import Data.Coerce
( coerce )
import Data.Kind
( Type, Constraint )
import GHC.Generics
( Generic, Generic1, Generically1(..) )
import GHC.TypeNats
( Nat )
( Generic, Generic1(..), Generically1(..) )
-- MetaBrush
import Math.Module
@ -24,8 +25,14 @@ import Math.Linear
-- | Differentiable mappings between spaces.
infixr 0 ~>
type (~>) :: Type -> Type -> Type
newtype a ~> b = D { runD :: a -> D a b }
deriving stock instance Functor ( D a ) => Functor ( (~>) a )
newtype u ~> v = D { runD :: u -> D u v }
deriving stock instance Functor ( D u ) => Functor ( (~>) u )
instance ( Applicative ( D u ), Module ( D u Double ) ( D u v ) ) => Module Double ( T ( u ~> v ) ) where
origin = T $ D \ _ -> origin
T ( D f ) ^+^ T ( D g ) = T $ D \ t -> f t ^+^ g t
T ( D f ) ^-^ T ( D g ) = T $ D \ t -> f t ^-^ g t
a *^ T ( D f ) = T $ D \ t -> pure a *^ f t
-- | @D ( ℝ n ) v@ is \( \mathbb{R}[x_1, \ldots, x_n]/(x_1, \ldots, x_n)^3 \otimes_\mathbb{R} v \)
type D :: Type -> Type -> Type
@ -36,22 +43,25 @@ type instance D ( ℝ 2 ) = Dℝ2
type instance D ( ℝ 3 ) = Dℝ3
newtype Dℝ0 v = D0 { v :: v }
deriving stock ( Show, Eq, Functor, Generic, Generic1 )
deriving stock ( Show, Eq, Functor, Foldable, Traversable, Generic, Generic1 )
deriving newtype ( Num, Fractional, Floating )
deriving Applicative
via Generically1 Dℝ0
data Dℝ1 v = D1 { v :: !v, dx :: !( T v ), ddx :: !( T v ) }
deriving stock ( Show, Eq, Functor, Generic, Generic1 )
data Dℝ1 v = D1 { v :: !v, df :: !( T v ), d²f :: !( T v ) }
deriving stock ( Eq, Functor, Foldable, Traversable, Generic, Generic1 )
deriving Applicative
via Generically1 Dℝ1
deriving stock instance ( Show v, Show ( T v ) ) => Show ( Dℝ1 v )
data Dℝ2 v = D2 { v :: !v, dx, dy :: !( T v ), ddx, dxdy, ddy :: !( T v ) }
deriving stock ( Show, Eq, Functor, Generic, Generic1 )
deriving stock ( Eq, Functor, Foldable, Traversable, Generic, Generic1 )
deriving Applicative
via Generically1 Dℝ2
deriving stock instance ( Show v, Show ( T v ) ) => Show ( Dℝ2 v )
data Dℝ3 v = D3 { v :: !v, dx, dy, dz :: !( T v ), ddx, dxdy, ddy, dxdz, dydz, ddz :: !( T v ) }
deriving stock ( Show, Eq, Functor, Generic, Generic1 )
deriving stock ( Eq, Functor, Foldable, Traversable, Generic, Generic1 )
deriving Applicative
via Generically1 Dℝ3
deriving stock instance ( Show v, Show ( T v ) ) => Show ( Dℝ3 v )
instance Num ( Dℝ1 Double ) where
(+) = liftA2 (+)
@ -110,29 +120,29 @@ instance Num ( Dℝ3 Double ) where
( T $ dz1 * dz2 + v1 * ddz2 + ddz1 * v2)
instance Module Double v => Module ( Dℝ0 Double ) ( Dℝ0 v ) where
(^+^) = liftA2 (^+^)
(^-^) = liftA2 (^-^)
origin = pure origin
(*^) = liftA2 (*^)
instance Module Double ( T v ) => Module ( Dℝ0 Double ) ( Dℝ0 v ) where
(^+^) = liftA2 ( coerce $ (^+^) @Double @( T v ) )
(^-^) = liftA2 ( coerce $ (^-^) @Double @( T v ) )
origin = pure ( coerce $ origin @Double @( T v ) )
(*^) = liftA2 ( coerce $ (*^) @Double @( T v ) )
instance Module Double v => Module ( Dℝ1 Double ) ( Dℝ1 v ) where
(^+^) = liftA2 (^+^)
(^-^) = liftA2 (^-^)
origin = pure origin
(*^) = liftA2 (*^)
instance Module Double ( T v ) => Module ( Dℝ1 Double ) ( Dℝ1 v ) where
(^+^) = liftA2 ( coerce $ (^+^) @Double @( T v ) )
(^-^) = liftA2 ( coerce $ (^-^) @Double @( T v ) )
origin = pure ( coerce $ origin @Double @( T v ) )
(*^) = liftA2 ( coerce $ (*^) @Double @( T v ) )
instance Module Double v => Module ( Dℝ2 Double ) ( Dℝ2 v ) where
(^+^) = liftA2 (^+^)
(^-^) = liftA2 (^-^)
origin = pure origin
(*^) = liftA2 (*^)
instance Module Double ( T v ) => Module ( Dℝ2 Double ) ( Dℝ2 v ) where
(^+^) = liftA2 ( coerce $ (^+^) @Double @( T v ) )
(^-^) = liftA2 ( coerce $ (^-^) @Double @( T v ) )
origin = pure ( coerce $ origin @Double @( T v ) )
(*^) = liftA2 ( coerce $ (*^) @Double @( T v ) )
instance Module Double v => Module ( Dℝ3 Double ) ( Dℝ3 v ) where
(^+^) = liftA2 (^+^)
(^-^) = liftA2 (^-^)
origin = pure origin
(*^) = liftA2 (*^)
instance Module Double ( T v ) => Module ( Dℝ3 Double ) ( Dℝ3 v ) where
(^+^) = liftA2 ( coerce $ (^+^) @Double @( T v ) )
(^-^) = liftA2 ( coerce $ (^-^) @Double @( T v ) )
origin = pure ( coerce $ origin @Double @( T v ) )
(*^) = liftA2 ( coerce $ (*^) @Double @( T v ) )
instance Fractional ( Dℝ1 Double ) where
(/) = error "I haven't yet defined (/) for Dℝ1"
@ -204,48 +214,67 @@ instance Floating ( Dℝ3 Double ) where
--------------------------------------------------------------------------------
uncurryD :: ( ℝ 1 ~> ℝ 1 ~> b ) -> ( ℝ 2 ~> b )
uncurryD ( D b ) = D \ ( ℝ2 t0 s0 ) ->
let !(D1 ( D b_t0 ) ( T ( D dbdt_t0 ) ) ( T ( D d2bdt2_t0 ) ) ) = b ( ℝ1 t0 )
!(D1 b_t0s0 dbds_t0s0 d2bds2_t0s0 ) = b_t0 ( ℝ1 s0 )
!(D1 dbdt_t0s0 d2bdtds_t0s0 _ ) = dbdt_t0 ( ℝ1 s0 )
!(D1 d2bdt2_t0s0 _ _ ) = d2bdt2_t0 ( ℝ1 s0 )
uncurryD ( D b ) = D \ ( ℝ2 t0 s0 ) -> uncurryD' ( b ( ℝ1 t0 ) ) ( ℝ1 s0 )
uncurryD' :: D ( ℝ 1 ) ( ℝ 1 ~> b ) -> ℝ 1 -> D ( ℝ 2 ) b
uncurryD' ( D1 ( D b_t0 ) ( T ( D dbdt_t0 ) ) ( T ( D d2bdt2_t0 ) ) ) ( ℝ1 s0 ) =
let !( D1 b_t0s0 dbds_t0s0 d2bds2_t0s0 ) = b_t0 ( ℝ1 s0 )
!( D1 dbdt_t0s0 d2bdtds_t0s0 _ ) = dbdt_t0 ( ℝ1 s0 )
!( D1 d2bdt2_t0s0 _ _ ) = d2bdt2_t0 ( ℝ1 s0 )
in D2 b_t0s0 ( T dbdt_t0s0 ) dbds_t0s0 ( T d2bdt2_t0s0 ) d2bdtds_t0s0 d2bds2_t0s0
--------------------------------------------------------------------------------
type Diffy :: Type -> Constraint
class ( Applicative ( D v )
, Traversable ( D v )
, Module Double ( T v ) )
=> Diffy v where
chain :: ( Module Double ( T w ) )
=> D ( ℝ 1 ) v -> D v w -> D ( ℝ 1 ) w
konst :: Module Double ( T w ) => w -> D v w
value :: D v w -> w
linear :: Module Double ( T w ) => ( v -> w ) -> ( v ~> w )
chainRule :: ( Diffy v, Module Double ( T w ) )
=> ( ( ℝ 1 ) ~> v )
-> ( v ~> w )
-> ( ( ℝ 1 ) ~> w )
chainRule ( D f ) ( D g ) =
=> ( ( ℝ 1 ) ~> v ) -> ( v ~> w ) -> ( ( ℝ 1 ) ~> w )
chainRule ( D df ) ( D dg ) =
D \ x ->
case f x of
df@( D1 { v = f_x } ) ->
chain df ( g f_x )
case df x of
df_x@( D1 { v = f_x } ) ->
chain df_x ( dg f_x )
-- | Recover the underlying function, discarding all infinitesimal information.
fun :: forall v w. Diffy v => ( v ~> w ) -> ( v -> w )
fun ( D f ) = value @v . f
fun ( D df ) = value @v . df
{-# INLINE fun #-}
type Diffy :: Type -> Constraint
class Diffy v where
chain :: ( Module Double ( T w ) )
=> Dℝ1 v -> D v w -> D ( ℝ 1 ) w
konst :: Module Double ( T w ) => w -> D v w
value :: D v w -> w
var :: ( Representable u, Diffy u ) => Fin ( Dim u ) -> ( u ~> Double )
var i = linear ( `index` i )
{-# INLINE var #-}
instance Diffy ( ℝ 0 ) where
chain _ ( D0 w ) = D1 w origin origin
{-# INLINE chain #-}
konst k = D0 k
{-# INLINE konst #-}
value ( D0 w ) = w
{-# INLINE value #-}
linear f = D \ _ -> D0 ( f ℝ0 )
{-# INLINE linear #-}
instance Diffy ( ℝ 1 ) where
chain ( D1 _ ( T ( ℝ1 x' ) ) ( T ( ℝ1 x'' ) ) ) ( D1 v g_x g_xx )
= D1 v
( x' *^ g_x )
( x'' *^ g_x ^+^ ( x' * x' ) *^ g_xx )
{-# INLINE chain #-}
konst k = D1 k origin origin
{-# INLINE konst #-}
value ( D1 { v } ) = v
{-# INLINE value #-}
linear f = D \ u -> D1 ( f u ) ( T $ f u ) origin
{-# INLINE linear #-}
instance Diffy ( ℝ 2 ) where
chain ( D1 _ ( T ( ℝ2 x' y' ) ) ( T ( ℝ2 x'' y'' ) ) ) ( D2 v g_x g_y g_xx g_xy g_yy )
@ -254,8 +283,16 @@ instance Diffy ( ℝ 2 ) where
( x'' *^ g_x ^+^ y'' *^ g_y
^+^ ( x' * x' ) *^ g_xx ^+^ ( y' * y' ) *^ g_yy
^+^ 2 *^ ( ( x' * y' ) *^ g_xy ) )
{-# INLINE chain #-}
konst k = D2 k origin origin origin origin origin
{-# INLINE konst #-}
value ( D2 { v } ) = v
{-# INLINE value #-}
linear f = D \ u ->
D2 ( f u )
( T $ f ( ℝ2 1 0 ) ) ( T $ f ( ℝ2 0 1 ) )
origin origin origin
{-# INLINE linear #-}
instance Diffy ( ℝ 3 ) where
chain ( D1 _ ( T ( ℝ3 x' y' z' ) ) ( T ( ℝ3 x'' y'' z'' ) ) )
@ -265,24 +302,13 @@ instance Diffy ( ℝ 3 ) where
( x'' *^ g_x ^+^ y'' *^ g_y ^+^ z'' *^ g_z
^+^ ( x' * x' ) *^ g_xx ^+^ ( y' * y' ) *^ g_yy ^+^ ( z' * z' ) *^ g_zz
^+^ 2 *^ ( ( x' * y' ) *^ g_xy ) ^+^ ( x' * z' ) *^ g_xz ^+^ ( y' * z' ) *^ g_yz )
{-# INLINE chain #-}
konst k = D3 k origin origin origin origin origin origin origin origin origin
{-# INLINE konst #-}
value ( D3 { v } ) = v
--------------------------------------------------------------------------------
type Var :: Nat -> Type -> Constraint
class Var n v where
var :: v ~> Double
instance Var 1 ( ℝ 1 ) where
var = D \ ( ℝ1 x ) -> D1 x ( T 1 ) ( T 0 )
instance Var 1 ( ℝ 2 ) where
var = D \ ( ℝ2 x _ ) -> D2 x ( T 1 ) ( T 0 ) ( T 0 ) ( T 0 ) ( T 0 )
instance Var 2 ( ℝ 2 ) where
var = D \ ( ℝ2 _ y ) -> D2 y ( T 0 ) ( T 1 ) ( T 0 ) ( T 0 ) ( T 0 )
instance Var 1 ( ℝ 3 ) where
var = D \ ( ℝ3 x _ _ ) -> D3 x ( T 1 ) ( T 0 ) ( T 0 ) ( T 0 ) ( T 0 ) ( T 0 ) ( T 0 ) ( T 0 ) ( T 0 )
instance Var 2 ( ℝ 3 ) where
var = D \ ( ℝ3 _ y _ ) -> D3 y ( T 0 ) ( T 1 ) ( T 0 ) ( T 0 ) ( T 0 ) ( T 0 ) ( T 0 ) ( T 0 ) ( T 0 )
instance Var 3 ( ℝ 3 ) where
var = D \ ( ℝ3 _ _ z ) -> D3 z ( T 0 ) ( T 0 ) ( T 1 ) ( T 0 ) ( T 0 ) ( T 0 ) ( T 0 ) ( T 0 ) ( T 0 )
{-# INLINE value #-}
linear f = D \ u ->
D3 ( f u )
( T $ f ( ℝ3 1 0 0 ) ) ( T $ f ( ℝ3 0 1 0 ) ) ( T $ f ( ℝ3 0 0 1 ) )
origin origin origin origin origin origin
{-# INLINE linear #-}

10
src/splines/Math/Module.hs
View file

@ -5,6 +5,7 @@
module Math.Module
( Module(..), lerp
, Inner(..)
, Interpolatable
, norm, squaredNorm, quadrance, distance
, proj, projC, closestPointOnSegment
, cross
@ -17,6 +18,8 @@ import Control.Applicative
( liftA2 )
import Control.Monad
( guard )
import Data.Kind
( Type, Constraint )
import Data.Monoid
( Ap(..), Sum(..) )
@ -109,6 +112,13 @@ closestPointOnSegment c ( Segment p0 p1 )
--------------------------------------------------------------------------------
-- | A convenient constraint synonym for types that support interpolation.
type Interpolatable :: Type -> Constraint
class ( Torsor ( T r ) r, Module Double ( T r ) ) => Interpolatable r
instance ( Torsor ( T r ) r, Module Double ( T r ) ) => Interpolatable r
--------------------------------------------------------------------------------
instance Num a => Module a ( Sum a ) where
origin = Sum 0

248
src/splines/Math/Roots.hs
View file

@ -3,7 +3,18 @@
{-# OPTIONS_GHC -Wno-name-shadowing #-}
module Math.Roots where
module Math.Roots
( -- * Quadratic solver
solveQuadratic
-- * Laguerre method
, roots, realRoots
, laguerre, eval, derivative
-- * Newton–Raphson
, newtonRaphson
)
where
-- base
import Control.Monad
@ -72,6 +83,9 @@ solveQuadratic a0 a1 a2
disc :: a
disc = a1 * a1 - 4 * a0 * a2
--------------------------------------------------------------------------------
-- Root finding using Laguerre's method
-- | Find real roots of a polynomial with real coefficients.
--
-- Coefficients are given in order of decreasing degree, e.g.:
@ -279,118 +293,140 @@ derivative p = do
pure p'
--------------------------------------------------------------------------------
data NewtonRaphson
= NewtonRaphson
{ f :: Double -> (# Double, Double #)
, maxIters :: Word
, min_x, max_x :: Double
, digits :: Int }
-- Newton–Raphson
-- Newton–Raphson implementation taken from Boost C++ library:
-- https://github.com/boostorg/math/blob/0dc6a70caa6bbec2b6ae25eede36c430f0ccae13/include/boost/math/tools/roots.hpp#L217
newtonRaphson :: NewtonRaphson
-> Double
-> Maybe Double
newtonRaphson ( NewtonRaphson { f, maxIters, min_x, max_x, digits } ) x0 =
go $
NewtonRaphsonState
{ f_x_prev = 0
, x = x0
, δ = maxDouble, δ1 = maxDouble
, iters = 0
, min_x, max_x
, min_f_x = 0, max_f_x = 0 }
{-# SPECIALISE
newtonRaphson
:: Word -> Int -> ( Double, Double ) -> ( Double -> ( Double, Double ) ) -> Double -> Maybe Double
#-}
{-# INLINEABLE newtonRaphson #-}
newtonRaphson :: ( RealFloat r, Show r )
=> Word -- ^ maximum number of iterations
-> Int -- ^ desired digits of precision
-> ( r, r ) -- ^ @(min_x, max_x)@.
-> ( r -> ( r, r ) ) -- ^ function and its derivative
-> r -- ^ initial guess
-> Maybe r
newtonRaphson maxIters digits ( min_x, max_x ) f x0 =
doNewtonRaphson f maxIters factor min_x max_x 0 0 0 x0 maxRealFloat maxRealFloat
where
factor = encodeFloat 1 ( 1 - digits )
go ( NewtonRaphsonState { f_x_prev, x, δ = δ1, δ1 = δ2, iters, min_x, max_x, min_f_x, max_f_x } )
= case f x of
(# f_x, f'_x #)
| f_x == 0
!factor = encodeFloat 1 ( 1 - digits )
doNewtonRaphson :: ( Fractional r, Ord r, Show r )
=> ( r -> (r, r) )
-> Word
-> r
-> r -> r
-> r -> r
-> r
-> r
-> r -> r
-> Maybe r
doNewtonRaphson f maxIters factor min_x max_x min_f_x max_f_x f_x_prev x δ1 δ2 =
go min_x max_x min_f_x max_f_x f_x_prev 0 x δ1 δ2
where
go min_x max_x min_f_x max_f_x f_x_prev !iters !x !δ1 !δ2 =
case f x of
( f_x, f'_x )
| f_x == 0
-> Just x
| ( new_x, δ, δ1 ) <- newtonRaphsonStep f min_x max_x f_x_prev x f_x f'_x δ1 δ2
-> if
| abs δ <= abs ( new_x * factor )
|| new_x == min_x || new_x == max_x
-> Just x
| δ <-
if | f'_x == 0
-> handleZeroDerivative f_x_prev x f_x δ1 min_x max_x
| otherwise
-> f_x / f'_x
, (# δ, δ1 #) <-
if | abs ( δ * δ ) > abs δ2
, let shift = if δ > 0 then 0.5 * ( x - min_x ) else 0.5 * ( x - max_x )
, δ <- if x /= 0 && ( abs shift > abs x )
then signum δ * abs x * 1.1
else shift
-> (# δ, 3 * δ #)
| otherwise
-> (# δ, δ1 #)
, let new_x = x - δ
, (# δ, new_x #) <-
if | new_x <= min_x
, δ <- 0.5 * ( x - min_x )
-> (# δ, x - δ #)
| new_x >= max_x
, δ <- 0.5 * ( x - max_x )
-> (# δ, x - δ #)
| otherwise
-> (# δ, new_x #)
-> if
| abs δ <= abs ( new_x * factor )
|| new_x == min_x || new_x == max_x
-> Just x
| iters >= maxIters
-> Nothing
| (# min_x, max_x, min_f_x, max_f_x #) <-
if δ > 0
then (# min_x, x, min_f_x, f_x #)
else (# x, max_x, f_x, max_f_x #)
-> if min_f_x * max_f_x > 0
then Nothing
else
go $ NewtonRaphsonState
{ f_x_prev = f_x, x = new_x
, δ, δ1
, iters = iters + 1
, min_x, max_x
, min_f_x, max_f_x
}
| iters >= maxIters
-> Nothing
| ( min_x, max_x, min_f_x, max_f_x ) <-
if δ > 0
then ( min_x, x, min_f_x, f_x )
else ( x, max_x, f_x, max_f_x )
-> if min_f_x * max_f_x > 0
then Nothing
else
go min_x max_x min_f_x max_f_x f_x ( iters + 1 ) new_x δ δ1
handleZeroDerivative :: Double
-> Double -> Double
-> Double
-> Double -> Double
-> Double
handleZeroDerivative f_x_prev x f_x δ min_x max_x
-- Handle zero derivative on first iteration.
| f_x_prev == 0
, x_prev <- if x <= 0.5 * ( min_x + max_x ) then max_x else min_x
, (# f_x_prev, _ #) <- f x_prev
, δ <- x_prev - x
= finish f_x_prev δ
| otherwise
= finish f_x_prev δ
newtonRaphsonStep :: ( Fractional r, Ord r, Show r )
=> ( r -> ( r, r ) )
-> r
-> r -> r -> r
-> r -> r
-> r -> r
-> ( r, r, r )
newtonRaphsonStep f min_x max_x f_x_prev x f_x f'_x δ1 δ2
| δ <-
if | f'_x == 0
-> zeroDerivativeStep f min_x max_x f_x_prev x f_x δ1
| otherwise
-> f_x / f'_x
, ( δ, δ1 ) <-
if | abs ( δ * δ ) > abs δ2
-> bisectStep min_x max_x x δ
| otherwise
-> ( δ, δ1 )
, let !new_x = x - δ
, ( δ, new_x ) <-
if | new_x <= min_x
, δ <- 0.5 * ( x - min_x )
-> ( δ, x - δ )
| new_x >= max_x
, δ <- 0.5 * ( x - max_x )
-> ( δ, x - δ )
| otherwise
-> ( δ, new_x )
= ( new_x, δ, δ1 )
where
finish f_x_prev δ
| signum f_x_prev * signum f_x < 0
= if δ < 0 then 0.5 * ( x - min_x ) else 0.5 * ( x - max_x )
| otherwise
= if δ < 0 then 0.5 * ( x - max_x ) else 0.5 * ( x - min_x )
-- | Loop state for the 'newtonRaphson' function.
data NewtonRaphsonState =
NewtonRaphsonState
{ f_x_prev :: !Double
, x :: !Double
, δ, δ1 :: !Double
, iters :: !Word
, min_x, max_x :: !Double
, min_f_x, max_f_x :: !Double }
maxDouble :: Double
maxDouble = encodeFloat m n
-- Handle \( f'(x_0) = 0 \).
zeroDerivativeStep :: ( Fractional r, Ord r, Show r )
=> ( r -> ( r, r ) )
-> r
-> r -> r
-> r
-> r -> r
-> r
zeroDerivativeStep f min_x max_x f_x_prev x f_x δ
-- Handle zero derivative on first iteration.
| f_x_prev == 0
, x_prev <- if x <= 0.5 * ( min_x + max_x ) then max_x else min_x
, ( f_x_prev, _ ) <- f x_prev
, δ <- x_prev - x
= finish f_x_prev δ
| otherwise
= finish f_x_prev δ
where
b = floatRadix ( 0 :: Double )
e = floatDigits ( 0 :: Double )
(_, e') = floatRange ( 0 :: Double )
finish f_x_prev δ
| signum f_x_prev * signum f_x < 0
= if δ < 0 then 0.5 * ( x - min_x ) else 0.5 * ( x - max_x )
| otherwise
= if δ < 0 then 0.5 * ( x - max_x ) else 0.5 * ( x - min_x )
-- Handle failure of convergence in the past two steps.
bisectStep :: ( Fractional r, Ord r, Show r ) => r -> r -> r -> r -> ( r, r )
bisectStep min_x max_x x δ
| let !shift = if δ > 0 then 0.5 * ( x - min_x ) else 0.5 * ( x - max_x )
, δ <- if x /= 0 && ( abs shift > abs x )
then signum δ * abs x * 1.1
else shift
= ( δ, 3 * δ )
{-# SPECIALISE maxRealFloat :: Float #-}
{-# SPECIALISE maxRealFloat :: Double #-}
maxRealFloat :: forall r. RealFloat r => r
maxRealFloat = encodeFloat m n
where
b = floatRadix @r 0
e = floatDigits @r 0
(_, e') = floatRange @r 0
m = b ^ e - 1
n = e' - e
{-
for some reason GHC isn't able to optimise maxRealFloat @Double into
maxDouble :: Double
maxDouble = D# 1.7976931348623157e308##
-}