mirror of
https://gitlab.com/sheaf/metabrush.git
synced 2024-11-23 15:34:06 +00:00
compute brush strokes using the envelope equation
This commit is contained in:
parent
43098dec01
commit
8fc5e6c9b8
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@ -36,6 +36,8 @@ common common
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>= 4.17 && < 4.18
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, acts
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^>= 0.3.1.0
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, code-page
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^>= 0.2.1
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, containers
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>= 0.6.0.1 && < 0.7
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, deepseq
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@ -112,6 +114,7 @@ common common
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common extras
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build-depends:
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directory
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>= 1.3.4.0 && < 1.4
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, filepath
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@ -180,6 +183,7 @@ library splines
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, Math.Module
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, Math.Orientation
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, Math.Roots
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, Debug.Utils
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build-depends:
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bifunctors
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@ -1,6 +1,4 @@
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{-# LANGUAGE BlockArguments #-}
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{-# LANGUAGE OverloadedStrings #-}
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{-# LANGUAGE TypeApplications #-}
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module Main
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( main )
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@ -14,6 +12,10 @@ import System.Exit
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import GHC.Conc
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( getNumProcessors, setNumCapabilities )
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-- code-page
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import System.IO.CodePage
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( withCP65001 )
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-- gi-gio
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import qualified GI.Gio as GIO
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@ -30,7 +32,7 @@ import MetaBrush.Application
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--------------------------------------------------------------------------------
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main :: IO ()
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main = do
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main = withCP65001 do
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procs <- getNumProcessors
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let
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@ -821,7 +821,7 @@ instance HandleAction MouseClick where
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changeText :: Text
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changeText = "Subdivide " <> loc
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pure ( UpdateDoc . UpdateDocumentTo $ HistoryChange {..} )
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-- Ignore double click event otherwise.
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_ -> pure Don'tModifyDoc
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@ -84,7 +84,6 @@ import Math.Linear
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import MetaBrush.Action
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( ActionOrigin(..) )
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import qualified MetaBrush.Asset.Brushes as Asset.Brushes
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( EllipseBrushFields, ellipse )
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import MetaBrush.Asset.Colours
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( getColours )
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import MetaBrush.Asset.Logo
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@ -161,6 +160,12 @@ runApplication application = do
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, strokeUnique = strokeUnique
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, strokeBrush = Just Asset.Brushes.ellipse
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, strokeSpline =
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-- Spline
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-- { splineStart = mkPoint ( ℝ2 -20 -20 ) 5
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-- , splineCurves = OpenCurves $ Seq.fromList
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-- [ LineTo { curveEnd = NextPoint ( mkPoint ( ℝ2 20 20 ) 5 ), curveData = invalidateCache undefined }
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-- ]
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-- }
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Spline
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{ splineStart = mkPoint ( ℝ2 10 -20 ) 2 1 0
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, splineCurves = OpenCurves $ Seq.fromList
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@ -176,6 +181,8 @@ runApplication application = do
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where
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mkPoint :: ℝ 2 -> Double -> Double -> Double -> PointData ( Record Asset.Brushes.EllipseBrushFields )
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mkPoint pt a b phi = PointData pt Normal ( MkR $ ℝ3 a b phi )
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-- mkPoint :: ℝ 2 -> Double -> PointData ( Record Asset.Brushes.CircleBrushFields )
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-- mkPoint pt r = PointData pt Normal ( MkR $ ℝ1 r )
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recomputeStrokesTVar <- STM.newTVarIO @Bool False
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documentRenderTVar <- STM.newTVarIO @( ( Int32, Int32 ) -> Cairo.Render () ) ( const $ pure () )
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@ -193,11 +200,11 @@ runApplication application = do
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maxHistorySizeTVar <- STM.newTVarIO @Int 1000
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fitParametersTVar <- STM.newTVarIO @FitParameters
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( FitParameters
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{ maxSubdiv = 6
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, nbSegments = 12
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, dist_tol = 5e-3
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, t_tol = 1e-4
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, maxIters = 100
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{ maxSubdiv = 2 --3 -- 6
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, nbSegments = 3 --6 -- 12
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, dist_tol = 0.1 -- 5e-3
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, t_tol = 0.1 -- 1e-4
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, maxIters = 2 -- 100
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}
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)
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@ -292,17 +292,18 @@ strokeRenderData fitParams
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{ inject
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, project = toUsedParams :: Record pointFields -> Record usedFields }
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-> do
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let embedUsedParams r = inject r brush_defaults
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let embedUsedParams = inject brush_defaults
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-- Compute the outline using the brush function.
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( outline, fitPts ) <-
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computeStrokeOutline @( T ( Record usedFields) ) @clo
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fitParams ( toUsedParams . brushParams ) ( fun brushFn . embedUsedParams ) spline
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computeStrokeOutline @clo fitParams
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( toUsedParams . brushParams ) embedUsedParams brushFn
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spline
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pure $
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StrokeWithOutlineRenderData
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{ strokeDataSpline = spline
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, strokeOutlineData = ( outline, fitPts )
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, strokeBrushFunction = fun brushFn . embedUsedParams . toUsedParams
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, strokeBrushFunction = fun brushFn . fun embedUsedParams . toUsedParams
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}
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_ -> pure $
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StrokeRenderData
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@ -604,7 +605,7 @@ drawFitPoint _ zoom ( FitPoint { fitPoint = ℝ2 x y } ) = do
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lift do
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Cairo.save
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Cairo.translate x y
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Cairo.arc 0 0 ( 2 / zoom ) 0 ( 2 * pi )
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Cairo.arc 0 0 ( 6 / zoom ) 0 ( 2 * pi ) -- 2 / zoom
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Cairo.setSourceRGBA r g b 0.8
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Cairo.fill
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Cairo.restore
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@ -621,7 +622,7 @@ drawFitPoint _ zoom ( FitTangent { fitPoint = ℝ2 x y, fitTangent = V2 tx ty }
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Cairo.translate x y
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Cairo.moveTo 0 0
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Cairo.lineTo ( 0.05 * tx ) ( 0.05 * ty )
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Cairo.setLineWidth ( 1 / zoom )
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Cairo.setLineWidth ( 4 / zoom ) -- 1 / zoom
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Cairo.setSourceRGBA r g b 0.8
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Cairo.stroke
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Cairo.arc 0 0 ( 2 / zoom ) 0 ( 2 * pi )
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@ -55,12 +55,12 @@ data Mode
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data ToolBar
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= ToolBar
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{ selectionTool, penTool, pathTool, brushTool, metaTool, debugTool
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:: !GTK.ToggleButton
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:: !GTK.ToggleButton
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}
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createToolBar :: Variables -> Colours -> GTK.Box -> IO ToolBar
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createToolBar ( Variables {..} ) colours toolBar = do
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widgetAddClass toolBar "toolBar"
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GTK.widgetSetValign toolBar GTK.AlignStart
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@ -3,10 +3,6 @@
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module MetaBrush.Asset.Brushes where
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-- base
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import Data.Coerce
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( coerce )
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-- containers
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import qualified Data.Sequence as Seq
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( fromList )
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@ -24,9 +20,9 @@ import qualified Data.HashMap.Strict as HashMap
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-- MetaBrush
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import Math.Bezier.Spline
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import Math.Linear
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( ℝ(..), T(..) )
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( ℝ(..), Fin(..) )
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import Math.Linear.Dual
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( D, type (~>)(..), Var(var), konst )
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( D, type (~>)(..), var, konst )
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import Math.Module
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( Module((^+^), (*^)) )
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import MetaBrush.Brush
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@ -86,17 +82,16 @@ circleBrush :: Record CircleBrushFields ~> Spline 'Closed () ( ℝ 2 )
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circleBrush =
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D \ params ->
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let r :: D ( Record CircleBrushFields ) Double
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r = runD ( var @1 ) params
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r = runD ( var ( Fin 1## ) ) params
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mkPt :: Double -> Double -> D ( Record CircleBrushFields ) ( ℝ 2 )
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mkPt ( kon -> x ) ( kon -> y )
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= fmap coerce
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$ ( x * r ) *^ e_x
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= ( x * r ) *^ e_x
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^+^ ( y * r ) *^ e_y
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in circleSpline @( Record CircleBrushFields ) mkPt
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where
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e_x, e_y :: D ( Record CircleBrushFields ) ( T ( ℝ 2 ) )
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e_x = pure $ T $ ℝ2 1 0
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e_y = pure $ T $ ℝ2 0 1
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e_x, e_y :: D ( Record CircleBrushFields ) ( ℝ 2 )
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e_x = pure $ ℝ2 1 0
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e_y = pure $ ℝ2 0 1
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kon = konst @( Record CircleBrushFields )
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@ -104,25 +99,17 @@ ellipseBrush :: Record EllipseBrushFields ~> Spline 'Closed () ( ℝ 2 )
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ellipseBrush =
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D \ params ->
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let a, b, phi :: D ( Record EllipseBrushFields ) Double
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a = runD ( var @1 ) params
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b = runD ( var @2 ) params
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phi = runD ( var @3 ) params
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a = runD ( var ( Fin 1## ) ) params
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b = runD ( var ( Fin 2## ) ) params
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phi = runD ( var ( Fin 3## ) ) params
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mkPt :: Double -> Double -> D ( Record EllipseBrushFields ) ( ℝ 2 )
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mkPt ( kon -> x ) ( kon -> y )
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= fmap coerce
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$ ( x * a * cos phi - y * b * sin phi ) *^ e_x
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= ( x * a * cos phi - y * b * sin phi ) *^ e_x
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^+^ ( y * b * cos phi + x * a * sin phi ) *^ e_y
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in circleSpline @( Record EllipseBrushFields ) mkPt
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where
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e_x, e_y :: D ( Record EllipseBrushFields ) ( T ( ℝ 2 ) )
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e_x = pure $ T $ ℝ2 1 0
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e_y = pure $ T $ ℝ2 0 1
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e_x, e_y :: D ( Record EllipseBrushFields ) ( ℝ 2 )
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e_x = pure $ ℝ2 1 0
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e_y = pure $ ℝ2 0 1
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kon = konst @( Record EllipseBrushFields )
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--ellipseArc :: ℝ 2 ~> ℝ 2
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--ellipseArc = brushStroke ( linear myPath ) ( uncurryD $ fmap bezier3 myBrush )
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--testing :: Double -> Double -> (# Double, T ( ℝ 2 ) #)
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--testing :: Double -> Double -> (# Double, T (ℝ 2) #)
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--testing t s = envelopeEquation ellipseArc t s
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@ -1,6 +1,7 @@
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{-# LANGUAGE AllowAmbiguousTypes #-}
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{-# LANGUAGE ScopedTypeVariables #-}
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{-# LANGUAGE UndecidableInstances #-}
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{-# LANGUAGE AllowAmbiguousTypes #-}
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{-# LANGUAGE QuantifiedConstraints #-}
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{-# LANGUAGE ScopedTypeVariables #-}
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{-# LANGUAGE UndecidableInstances #-}
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module MetaBrush.Brush
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( Brush(..), SomeBrush(..), BrushFunction
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@ -38,12 +39,17 @@ import qualified Data.Text as Text
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-- MetaBrush
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import Math.Linear
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( ℝ, Representable )
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import Math.Linear.Dual
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( Diffy )
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import Math.Module
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( Interpolatable )
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import Math.Bezier.Spline
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( SplineType(Closed), SplinePts)
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import MetaBrush.Records
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( KnownSymbols, Length, Record, WithParams )
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import MetaBrush.Serialisable
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( Serialisable )
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--------------------------------------------------------------------------------
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@ -49,7 +49,7 @@ import Math.Bezier.Spline
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import Math.Bezier.Stroke
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( CachedStroke(..), invalidateCache )
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import Math.Module
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( lerp, quadrance, closestPointOnSegment )
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( Interpolatable, lerp, quadrance, closestPointOnSegment )
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import Math.Linear
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( Segment(..), ℝ(..), T(..) )
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import MetaBrush.Document
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@ -57,8 +57,6 @@ import MetaBrush.Document
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, PointData(..), DiffPointData(..)
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, coords, _strokeSpline
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)
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import MetaBrush.Records
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( Interpolatable )
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--------------------------------------------------------------------------------
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@ -54,13 +54,6 @@ import Math.Module
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--------------------------------------------------------------------------------
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-- | A convenient constraint synonym for types that support interpolation.
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type Interpolatable :: Type -> Constraint
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class ( Torsor ( T r ) r, Module Double ( T r ) ) => Interpolatable r
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instance ( Torsor ( T r ) r, Module Double ( T r ) ) => Interpolatable r
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--------------------------------------------------------------------------------
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-- | A function from a given record type, with provided default values
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-- that can be overridden.
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type WithParams :: [ Symbol ] -> Type -> Type
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@ -123,9 +116,11 @@ instance ( Torsor ( T ( ℝ ( Length ks ) ) ) ( ℝ ( Length ks ) )
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=> Torsor ( T ( Record ks ) ) ( Record ks ) where
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MkR g --> MkR a = T $ MkR $ unT $ g --> a
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type instance D ( Record ks ) = D ( ℝ ( Length ks ) )
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deriving newtype
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instance Representable ( ℝ ( Length ks ) )
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=> Representable ( Record ks )
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deriving newtype instance Var n ( ℝ ( Length ks ) ) => Var n ( Record ks )
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type instance D ( Record ks ) = D ( ℝ ( Length ks ) )
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deriving newtype instance Diffy ( ℝ ( Length ks ) ) => Diffy ( Record ks )
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--------------------------------------------------------------------------------
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@ -154,31 +149,35 @@ intersect :: forall r1 r2 l1 l2
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, KnownSymbols r1, KnownSymbols r2
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, l1 ~ Length r1, l2 ~ Length r2
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, Representable ( ℝ l1 ), Representable ( ℝ l2 )
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, Interpolatable ( Record r1 )
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, Interpolatable ( Record r1 ), Diffy ( ℝ l2 )
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)
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=> Intersection r1 r2
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intersect
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-- Shortcut when the two rows are equal.
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| Just Refl <- eqT @r1 @r2
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, Refl <- ( unsafeCoerce Refl :: r1 :~: Intersect r1 r2 )
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= Intersection { project = id, inject = const }
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= Intersection { project = id, inject = \ _ -> linear id }
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| otherwise
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= doIntersection @r1 @r2 \ ( _ :: Proxy# r1r2 ) r1_idxs r2_idxs ->
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let
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project :: Record r1 -> Record r1r2
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project = \ ( MkR r1 ) -> MkR $ projection ( (!) r1_idxs ) r1
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inject :: Record r1r2 -> Record r2 -> Record r2
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inject = \ ( MkR r1r2 ) ( MkR r2 ) -> MkR $ injection ( find eqFin r2_idxs ) r1r2 r2
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inject :: Record r2 -> Record r1r2 ~> Record r2
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inject = \ ( MkR r2 ) -> linear \ ( MkR r1r2 ) -> MkR $ injection ( find eqFin r2_idxs ) r1r2 r2
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in Intersection { project, inject }
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data Intersection r1 r2 where
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Intersection
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:: forall r1r2 r1 r2
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. ( KnownSymbols r1r2, Representable ( ℝ ( Length r1r2 ) )
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:: forall r1r2 r1 r2 l12
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. ( l12 ~ Length r1r2
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, KnownSymbols r1r2
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, Representable ( ℝ l12 )
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, Diffy ( ℝ l12 )
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, Interpolatable ( Record r1r2 ) )
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=> { project :: Record r1 -> Record r1r2
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, inject :: Record r1r2 -> Record r2 -> Record r2
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=> { project :: Record r1 -> Record r1r2
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, inject :: Record r2 -> Record r1r2 ~> Record r2
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-- ^ overrides the components of the first record with the second
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} -> Intersection r1 r2
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{-# INLINE doIntersection #-}
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|
@ -190,7 +189,7 @@ doIntersection
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)
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=> ( forall r1r2 l12.
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( r1r2 ~ Intersect r1 r2, l12 ~ Length r1r2
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, Representable ( ℝ l12 ), Interpolatable ( ℝ l12 )
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, Representable ( ℝ l12 ), Diffy ( ℝ l12 ), Interpolatable ( ℝ l12 )
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, KnownSymbols r1r2, Representable ( ℝ ( Length r1r2 ) )
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)
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=> Proxy# r1r2 -> Vec l12 ( Fin l1 ) -> Vec l12 ( Fin l2 ) -> kont )
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|
|
26
src/splines/Debug/Utils.hs
Normal file
26
src/splines/Debug/Utils.hs
Normal file
|
@ -0,0 +1,26 @@
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module Debug.Utils ( trace ) where
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|
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-- base
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||||
import System.IO
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( BufferMode(..), hSetBuffering, hFlush, hPutStrLn, stdout )
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import System.IO.Unsafe
|
||||
( unsafePerformIO )
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|
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-- deepseq
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import Control.DeepSeq
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( force )
|
||||
|
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-- code-page
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import System.IO.CodePage
|
||||
( withCP65001 )
|
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|
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--------------------------------------------------------------------------------
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|
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trace :: String -> a -> a
|
||||
trace ( force -> !str ) expr =
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unsafePerformIO $ withCP65001 do
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hSetBuffering stdout ( BlockBuffering $ Just 50 )
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hPutStrLn stdout str
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hFlush stdout
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return expr
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@ -120,7 +120,7 @@ data FitPoint
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-- including the meaning of \( \texttt{t_tol} \) and \( \texttt{maxIters} \).
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fitSpline
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:: FitParameters
|
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-> ( Double -> ( ℝ 2, T ( ℝ 2 ) ) ) -- ^ curve \( t \mapsto ( C(t), C'(t) ) \) to fit
|
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-> ( ℝ 1 -> ( ℝ 2, T ( ℝ 2 ) ) ) -- ^ curve \( t \mapsto ( C(t), C'(t) ) \) to fit
|
||||
-> ( SplinePts Open, Seq FitPoint )
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fitSpline ( FitParameters {..} ) = go 0
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where
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||||
|
@ -128,16 +128,16 @@ fitSpline ( FitParameters {..} ) = go 0
|
|||
dt = recip ( fromIntegral nbSegments )
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go
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||||
:: Int
|
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-> ( Double -> ( ℝ 2, T ( ℝ 2 ) ) )
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||||
-> ( ℝ 1 -> ( ℝ 2, T ( ℝ 2 ) ) )
|
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-> ( SplinePts Open, Seq FitPoint )
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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 ] ]
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(p, tp) = curveFn $ ℝ1 0
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(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 )
|
||||
|
|
|
@ -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
|
||||
|
|
|
@ -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
|
||||
|
|
|
@ -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 #-}
|
||||
|
|
|
@ -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
|
||||
|
|
|
@ -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##
|
||||
|
||||
-}
|
||||
|
|
Loading…
Reference in a new issue