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continue interval arithmetic integration
This commit is contained in:
sheaf
parent
a4e9c1cf32
commit
09c1bdd948
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@ -76,6 +76,7 @@ common common
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PatternSynonyms
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RankNTypes
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RecordWildCards
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RoleAnnotations
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StandaloneDeriving
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StandaloneKindSignatures
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TupleSections
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@ -18,7 +18,7 @@ import Data.Functor.Const
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import Data.Functor.Identity
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( Identity(..) )
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import Data.Kind
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( Constraint )
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( Type, Constraint )
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import Data.Monoid
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( Ap(..) )
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import Data.Semigroup
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@ -87,7 +87,8 @@ instance SplineTypeI Open where
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instance SplineTypeI Closed where
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ssplineType = SClosed
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data family NextPoint ( clo :: SplineType ) ptData
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type NextPoint :: SplineType -> Type -> Type
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data family NextPoint clo
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newtype instance NextPoint Open ptData = NextPoint { nextPoint :: ptData }
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deriving stock ( Show, Generic, Generic1, Functor, Foldable, Traversable )
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deriving anyclass ( NFData, NFData1 )
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@ -111,7 +112,8 @@ toNextPoint pt = case ssplineType @clo of
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SOpen -> NextPoint pt
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SClosed -> BackToStart
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data Curve ( clo :: SplineType ) crvData ptData
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type Curve :: SplineType -> Type -> Type -> Type
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data Curve clo crvData ptData
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= LineTo
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{ curveEnd :: !( NextPoint clo ptData )
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, curveData :: !crvData
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@ -157,7 +159,8 @@ openCurveStart ( LineTo ( NextPoint p ) _ ) = p
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openCurveStart ( Bezier2To p _ _ ) = p
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openCurveStart ( Bezier3To p _ _ _ ) = p
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data family Curves ( clo :: SplineType ) crvData ptData
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type Curves :: SplineType -> Type -> Type -> Type
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data family Curves clo
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newtype instance Curves Open crvData ptData
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= OpenCurves { openCurves :: Seq ( Curve Open crvData ptData ) }
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@ -1,8 +1,7 @@
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{-# LANGUAGE AllowAmbiguousTypes #-}
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{-# LANGUAGE QuantifiedConstraints #-}
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{-# LANGUAGE ScopedTypeVariables #-}
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{-# LANGUAGE DuplicateRecordFields #-}
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{-# LANGUAGE UndecidableInstances #-}
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module Math.Bezier.Stroke
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( Offset(..)
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@ -32,11 +31,13 @@ import Control.Monad.ST
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import Data.Bifunctor
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( Bifunctor(bimap) )
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import Data.Coerce
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( coerce )
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( Coercible, coerce )
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import Data.Foldable
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( for_ )
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import Data.Functor.Identity
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( Identity(..) )
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import Data.Kind
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( Type )
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import Data.List.NonEmpty
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( unzip )
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import Data.Maybe
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@ -434,13 +435,16 @@ outlineFunction ptParams toBrushParams brushFromParams sp0 crv =
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case crv of
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LineTo { curveEnd = NextPoint sp1 }
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| let seg = Segment sp0 sp1
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-> ( line ( fmap coords seg ), line ( fmap ptParams seg ) )
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-> ( line @Point ( fmap coords seg )
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, line @Point ( fmap ptParams seg ) )
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Bezier2To { controlPoint = sp1, curveEnd = NextPoint sp2 }
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| let bez2 = Quadratic.Bezier sp0 sp1 sp2
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-> ( bezier2 ( fmap coords bez2 ), bezier2 ( fmap ptParams bez2 ) )
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-> ( bezier2 @Point ( fmap coords bez2 )
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, bezier2 @Point ( fmap ptParams bez2 ) )
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Bezier3To { controlPoint1 = sp1, controlPoint2 = sp2, curveEnd = NextPoint sp3 }
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| let bez3 = Cubic.Bezier sp0 sp1 sp2 sp3
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-> ( bezier3 ( fmap coords bez3 ), bezier3 ( fmap ptParams bez3 ) )
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-> ( bezier3 @Point ( fmap coords bez3 )
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, bezier3 @Point ( fmap ptParams bez3 ) )
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fwdBwd :: OutlineFn
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fwdBwd t
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@ -449,8 +453,12 @@ outlineFunction ptParams toBrushParams brushFromParams sp0 crv =
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-- , ( offset bwdOffset • path_t, -1 *^ path'_t ) )
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where
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curves :: Seq ( ℝ 1 -> StrokeDatum )
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curves = brushStrokeData path ( usedParams `chainRule` toBrushParams ) brushFromParams t
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curves :: Seq ( ℝ 1 -> StrokeDatum Point )
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curves = brushStrokeData @Point @brushParams
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path
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( usedParams `chainRule` toBrushParams )
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brushFromParams
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t
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fwdOffset = withTangent path'_t brush_t
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bwdOffset = withTangent ( -1 *^ path'_t ) brush_t
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@ -771,7 +779,7 @@ brushStroke ( D1 p dpdt d2pdt2 ) ( D2 b dbdt dbds d2bdt2 d2bdtds d2bds2 ) =
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-- | The envelope equation
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--
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-- \[ E = \frac{\partial c}{\partial t} \times \frac{\partial c}{\partial s} = 0, ]
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-- \[ E = \frac{\partial c}{\partial t} \times \frac{\partial c}{\partial s} = 0, \]
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--
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-- together with the total derivative
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--
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@ -782,12 +790,13 @@ brushStroke ( D1 p dpdt d2pdt2 ) ( D2 b dbdt dbds d2bdt2 d2bdtds d2bds2 ) =
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-- \[ \frac{\partial E}{\partial s}, \qquad \frac{\partial E}{\partial s}. \]
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--
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-- NB: if \( \frac{\partial E}{\partial s} \) is zero, the total derivative is ill-defined.
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envelopeEquation :: ( D ( i ( ℝ 2 ) ) ~ D ( ℝ 2 )
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, Cross ( i Double ) ( T ( i ( ℝ 2 ) ) )
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, Fractional ( i Double )
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envelopeEquation :: forall i
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. ( D ( I i ( ℝ 2 ) ) ~ D ( ℝ 2 )
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, Cross ( I i Double ) ( T ( I i ( ℝ 2 ) ) )
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, Fractional ( I i Double )
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)
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=> D ( i ( ℝ 2 ) ) ( i ( ℝ 2 ) )
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-> ( i Double, T ( i ( ℝ 2 ) ), i Double, i Double )
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=> D ( I i ( ℝ 2 ) ) ( I i ( ℝ 2 ) )
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-> ( I i Double, T ( I i ( ℝ 2 ) ), I i Double, I i Double )
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envelopeEquation ( D2 _ dcdt dcds d2cdt2 d2cdtds d2cds2 ) =
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let ee = dcdt `cross` dcds
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dEdt = d2cdt2 `cross` dcds + dcdt `cross` d2cdtds
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@ -802,30 +811,47 @@ envelopeEquation ( D2 _ dcdt dcds d2cdt2 d2cdtds d2cds2 ) =
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-- ∂E/∂s dc/dt = ∂E/∂s ∂c/∂t - ∂E/∂t ∂c/∂s.
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-- | Linear interpolation, as a differentiable function.
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line :: forall b. ( Module Double ( T b ), Torsor ( T b ) b )
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=> Segment b -> ℝ 1 ~> b
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line ( Segment a b ) = D \ ( ℝ1 t ) ->
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line :: forall i b
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. ( Module ( I i Double ) ( T b ), Torsor ( T b ) b
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, D ( I i ( ℝ 1 ) ) ~ D ( ℝ 1 )
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, Coercible ( I i ( ℝ 1 ) ) ( I i Double )
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)
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=> Segment b -> I i ( ℝ 1 ) ~> b
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line ( Segment a b ) = D \ ( coerce -> t ) ->
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D1 ( lerp @( T b ) t a b )
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( a --> b )
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origin
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-- | A quadratic Bézier curve, as a differentiable function.
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bezier2 :: forall b. ( Module Double ( T b ), Torsor ( T b ) b )
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=> Quadratic.Bezier b -> ℝ 1 ~> b
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bezier2 bez = D \ ( ℝ1 t ) ->
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bezier2 :: forall i b
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. ( Module ( I i Double ) ( T b ), Torsor ( T b ) b
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, D ( I i ( ℝ 1 ) ) ~ D ( ℝ 1 )
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, Coercible ( I i ( ℝ 1 ) ) ( I i Double )
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)
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=> Quadratic.Bezier b -> I i ( ℝ 1 ) ~> b
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bezier2 bez = D \ ( coerce -> t ) ->
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D1 ( Quadratic.bezier @( T b ) bez t )
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( Quadratic.bezier' bez t )
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( Quadratic.bezier'' bez )
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-- | A cubic Bézier curve, as a differentiable function.
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bezier3 :: forall b. ( Module Double ( T b ), Torsor ( T b ) b )
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=> Cubic.Bezier b -> ℝ 1 ~> b
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bezier3 bez = D \ ( ℝ1 t ) ->
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bezier3 :: forall i b
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. ( Module ( I i Double ) ( T b ), Torsor ( T b ) b
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, D ( I i ( ℝ 1 ) ) ~ D ( ℝ 1 )
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, Coercible ( I i ( ℝ 1 ) ) ( I i Double )
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)
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=> Cubic.Bezier b -> I i ( ℝ 1 ) ~> b
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bezier3 bez = D \ ( coerce -> t ) ->
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D1 ( Cubic.bezier @( T b ) bez t )
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( Cubic.bezier' bez t )
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( Cubic.bezier'' bez t )
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splineCurveFns :: SplinePts Closed -> Seq ( ℝ 1 ~> ℝ 2 )
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splineCurveFns :: forall i
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. ( D ( I i ( ℝ 1 ) ) ~ D ( ℝ 1 )
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, Module ( I i Double ) ( T ( I i ( ℝ 2 ) ) )
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, Torsor ( T ( I i ( ℝ 2 ) ) ) ( I i ( ℝ 2 ) )
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, Coercible ( I i ( ℝ 1 ) ) ( I i Double ) )
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=> Spline Closed () ( I i ( ℝ 2 ) ) -> Seq ( I i ( ℝ 1 ) ~> I i ( ℝ 2 ) )
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splineCurveFns spls
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= runIdentity
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. bifoldSpline
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@ -834,21 +860,23 @@ splineCurveFns spls
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. adjustSplineType @Open
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$ spls
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where
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curveFn :: ℝ 2 -> Curve Open () ( ℝ 2 ) -> ( ℝ 1 ~> ℝ 2 )
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curveFn :: I i ( ℝ 2 )
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-> Curve Open () ( I i ( ℝ 2 ) )
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-> ( I i ( ℝ 1 ) ~> I i ( ℝ 2 ) )
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curveFn p0 = \case
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LineTo { curveEnd = NextPoint p1 }
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-> line $ Segment p0 p1
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-> line @i $ Segment p0 p1
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Bezier2To { controlPoint = p1, curveEnd = NextPoint p2 }
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-> bezier2 $ Quadratic.Bezier p0 p1 p2
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-> bezier2 @i $ Quadratic.Bezier p0 p1 p2
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Bezier3To { controlPoint1 = p1, controlPoint2 = p2, curveEnd = NextPoint p3 }
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-> bezier3 $ Cubic.Bezier p0 p1 p2 p3
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-> bezier3 @i $ Cubic.Bezier p0 p1 p2 p3
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-- | Solve the envelope equations at a given point \( t = t_0 \), to find
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-- \( s_0 \) such that \( c(t_0, s_0) \) is on the envelope of the brush stroke.
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solveEnvelopeEquations :: ℝ 2
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-> T ( ℝ 2 )
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-> ( Offset, Offset )
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-> Seq ( ℝ 1 -> StrokeDatum )
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-> Seq ( ℝ 1 -> StrokeDatum Point )
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-> ( ( ℝ 2, T ( ℝ 2 ) ), ( ℝ 2, T ( ℝ 2 ) ) )
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solveEnvelopeEquations path_t path'_t ( fwdOffset, bwdOffset ) strokeData
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= ( fwdSol, ( bwdPt, -1 *^ bwdTgt ) )
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@ -909,7 +937,7 @@ solveEnvelopeEquations path_t path'_t ( fwdOffset, bwdOffset ) strokeData
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| otherwise
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= i + 1
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sol :: Double -> ( ℝ 1 -> StrokeDatum ) -> ( Bool, ℝ 1, ℝ 2, T ( ℝ 2 ) )
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sol :: Double -> ( ℝ 1 -> StrokeDatum Point ) -> ( Bool, ℝ 1, ℝ 2, T ( ℝ 2 ) )
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sol initialGuess f =
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let (good, s) = case newtonRaphson maxIters precision domain ( eqn f ) initialGuess of
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Nothing -> ( False, initialGuess )
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@ -937,7 +965,7 @@ solveEnvelopeEquations path_t path'_t ( fwdOffset, bwdOffset ) strokeData
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-- ] )
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( good, ℝ1 s, value @Double @( ℝ 2 ) dstroke, totDeriv )
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eqn :: ( ℝ 1 -> StrokeDatum ) -> ( Double -> ( Double, Double ) )
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eqn :: ( ℝ 1 -> StrokeDatum Point ) -> ( Double -> ( Double, Double ) )
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eqn f s =
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case f ( ℝ1 s ) of
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StrokeDatum { ee, 𝛿E𝛿s } ->
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@ -956,36 +984,47 @@ instance Applicative ZipSeq where
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pure _ = error "only use Applicative ZipSeq with non-empty Traversable functors"
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liftA2 f ( ZipSeq xs ) ( ZipSeq ys ) = ZipSeq ( Seq.zipWith f xs ys )
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brushStrokeData :: forall brushParams
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. ( Diffy Double brushParams, Show brushParams )
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=> ( ℝ 1 ~> ℝ 2 ) -- ^ path
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-> ( ℝ 1 ~> brushParams ) -- ^ brush parameters
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-> ( brushParams ~> SplinePts Closed ) -- ^ brush from parameters
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-> ( ℝ 1 -> Seq ( ℝ 1 -> StrokeDatum ) )
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brushStrokeData :: forall i brushParams
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. ( Diffy ( I i Double ) ( I i brushParams )
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, Fractional ( I i Double )
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, D ( I i ( ℝ 1 ) ) ~ D ( ℝ 1 )
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, D ( I i ( ℝ 2 ) ) ~ D ( ℝ 2 )
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, Cross ( I i Double ) ( T ( I i ( ℝ 2 ) ) )
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, Torsor ( T ( I i ( ℝ 2 ) ) ) ( I i ( ℝ 2 ) )
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, Coercible ( I i ( ℝ 1 ) ) ( I i Double )
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, Show brushParams
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)
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=> ( I i ( ℝ 1 ) ~> I i ( ℝ 2 ) )
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-- ^ path
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-> ( I i ( ℝ 1 ) ~> I i brushParams )
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-- ^ brush parameters
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-> ( I i brushParams ~> Spline Closed () ( I i ( ℝ 2 ) ) )
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-- ^ brush from parameters
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-> ( I i ( ℝ 1 ) -> Seq ( I i ( ℝ 1 ) -> StrokeDatum i ) )
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brushStrokeData path params brush =
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\ t ->
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let
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dpath_t :: D ( ℝ 1 ) ( ℝ 2 )
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dpath_t :: D ( I i ( ℝ 1 ) ) ( I i ( ℝ 2 ) )
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!dpath_t = runD path t
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dparams_t :: D ( ℝ 1 ) brushParams
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dparams_t :: D ( I i ( ℝ 1 ) ) ( I i brushParams )
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!dparams_t@( D1 { v = params_t } ) = runD params t
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dbrush_params :: D brushParams ( SplinePts Closed )
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dbrush_params :: D ( I i brushParams ) ( Spline Closed () ( I i ( ℝ 2 ) ) )
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!dbrush_params = runD brush params_t
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splines :: Seq ( D brushParams ( ℝ 1 ~> ℝ 2 ) )
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!splines = getZipSeq $ traverse ( ZipSeq . splineCurveFns ) dbrush_params
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dbrushes_t :: Seq ( ℝ 1 -> D ( ℝ 2 ) ( ℝ 2 ) )
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splines :: Seq ( D ( I i brushParams ) ( I i ( ℝ 1 ) ~> I i ( ℝ 2 ) ) )
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!splines = getZipSeq $ traverse ( ZipSeq . splineCurveFns @i ) dbrush_params
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dbrushes_t :: Seq ( I i ( ℝ 1 ) -> D ( I i ( ℝ 2 ) ) ( I i ( ℝ 2 ) ) )
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!dbrushes_t = force $ fmap ( uncurryD . ( dparams_t `chain` ) ) splines
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in fmap ( mkStrokeDatum dpath_t ) dbrushes_t
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where
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mkStrokeDatum :: D ( ℝ 1 ) ( ℝ 2 )
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-> ( ℝ 1 -> D ( ℝ 2 ) ( ℝ 2 ) )
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-> ( ℝ 1 -> StrokeDatum )
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mkStrokeDatum :: D ( I i ( ℝ 1 ) ) ( I i ( ℝ 2 ) )
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-> ( I i ( ℝ 1 ) -> D ( I i ( ℝ 2 ) ) ( I i ( ℝ 2 ) ) )
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-> ( I i ( ℝ 1 ) -> StrokeDatum i )
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mkStrokeDatum dpath_t dbrush_t s =
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let dbrush_t_s = dbrush_t s
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dstroke@( D2 _c _𝛿c𝛿t _𝛿c𝛿s _ _ _ ) = brushStroke dpath_t dbrush_t_s
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( ee, dcdt, 𝛿E𝛿t, 𝛿E𝛿s ) = coerce $ envelopeEquation @Identity $ coerce dstroke
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( ee, dcdt, 𝛿E𝛿t, 𝛿E𝛿s ) = envelopeEquation @i dstroke
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in -- trace
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-- ( unlines
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-- [ "envelopeEquation:"
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@ -1004,34 +1043,46 @@ brushStrokeData path params brush =
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, dstroke
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, ee, dcdt, 𝛿E𝛿t, 𝛿E𝛿s }
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-- | The value and derivative of a brush stroke at a given coordinate
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-- \( (t_0, s_0) \), together with the value of the envelope equation at that
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-- point.
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data StrokeDatum
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data StrokeDatum i
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= StrokeDatum
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{ -- | Path \( p(t_0) \) (with its 1st and 2nd derivatives).
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dpath :: D ( ℝ 1 ) ( ℝ 2 )
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dpath :: D ( I i ( ℝ 1 ) ) ( I i ( ℝ 2 ) )
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-- | Brush shape \( b(t_0, s_0) \) (with its 1st and 2nd derivatives).
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, dbrush :: D ( ℝ 2 ) ( ℝ 2 )
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, dbrush :: D ( I i ( ℝ 2 ) ) ( I i ( ℝ 2 ) )
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-- Everything below can be computed in terms of the first two fields.
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-- | Stroke \( c(t_0,s_0) = p(t_0) + b(t_0,s_0) \) (with its 1st and 2nd derivatives).
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, dstroke :: D ( ℝ 2 ) ( ℝ 2 )
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, dstroke :: D ( I i ( ℝ 2 ) ) ( I i ( ℝ 2 ) )
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-- | Envelope
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--
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-- \[ E(t_0,s_0) = \left ( \frac{\partial c}{\partial t} \times \frac{\partial c}{\partial s} \right )_{(t_0,s_0)}. \]
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, ee :: Double
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, ee :: I i Double
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-- | \( \left ( \frac{\partial E}{\partial s} \right )_{(t_0,s_0)}. \)
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, 𝛿E𝛿s :: Double
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, 𝛿E𝛿s :: I i Double
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-- | \( \left ( \frac{\partial E}{\partial t} \right )_{(t_0,s_0)}. \)
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, 𝛿E𝛿t :: Double
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, 𝛿E𝛿t :: I i Double
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-- | Total derivative
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--
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-- \[ \left ( \frac{\mathrm{d} c}{\mathrm{d} t} \right )_{(t_0,s_0)}. \]
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--
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-- This is ill-defined when \( \frac{\partial E}{\partial s} = 0 \).
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, dcdt :: T ( ℝ 2 )
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, dcdt :: T ( I i ( ℝ 2 ) )
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}
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deriving stock Show
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||||
deriving stock instance Show ( StrokeDatum Point )
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deriving stock instance Show ( StrokeDatum Interval )
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||||
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||||
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-- Handling points and intervals uniformly.
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data Extent = Point | Interval
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type I :: Extent -> Type -> Type
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type family I i a where
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I Point a = a
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I Interval a = 𝕀 a
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|
|
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|
@ -11,8 +11,6 @@ import Control.Applicative
|
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( liftA2 )
|
||||
import Data.Coerce
|
||||
( coerce )
|
||||
import Data.Functor.Identity
|
||||
( Identity(..) )
|
||||
import Data.Kind
|
||||
( Type, Constraint )
|
||||
import GHC.Generics
|
||||
|
|
@ -45,7 +43,6 @@ type instance D ( ℝ 1 ) = Dℝ1
|
|||
type instance D ( ℝ 2 ) = Dℝ2
|
||||
type instance D ( ℝ 3 ) = Dℝ3
|
||||
|
||||
type instance D ( Identity a ) = D a
|
||||
type instance D ( 𝕀 u ) = D u
|
||||
|
||||
newtype Dℝ0 v = D0 { v :: v }
|
||||
|
|
|
|||
|
|
@ -18,10 +18,6 @@ import Control.Applicative
|
|||
( liftA2 )
|
||||
import Control.Monad
|
||||
( guard )
|
||||
import Data.Coerce
|
||||
( coerce )
|
||||
import Data.Functor.Identity
|
||||
( Identity(..) )
|
||||
import Data.Kind
|
||||
( Type, Constraint )
|
||||
import Data.Monoid
|
||||
|
|
@ -176,16 +172,6 @@ instance Inner Double ( T ( ℝ 2 ) ) where
|
|||
instance Cross Double ( T ( ℝ 2 ) ) where
|
||||
cross ( V2 x1 y1 ) ( V2 x2 y2 ) = x1 * y2 - x2 * y1
|
||||
|
||||
instance Module r ( T m ) => Module ( Identity r ) ( T ( Identity m ) ) where
|
||||
origin = coerce $ origin @r @( T m )
|
||||
(^+^) = coerce $ (^+^) @r @( T m )
|
||||
(^-^) = coerce $ (^-^) @r @( T m )
|
||||
(*^) = coerce $ (*^) @r @( T m )
|
||||
instance Inner r ( T m ) => Inner ( Identity r ) ( T ( Identity m ) ) where
|
||||
(^.^) = coerce $ (^.^) @r @( T m )
|
||||
instance Cross r ( T m ) => Cross ( Identity r ) ( T ( Identity m ) ) where
|
||||
cross = coerce $ cross @r @( T m )
|
||||
|
||||
-- | Compute whether two vectors point in the same direction,
|
||||
-- that is, whether each vector is a (strictly) positive multiple of the other.
|
||||
--
|
||||
|
|
|
|||
Loading…
Reference in a new issue