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improve D to use Nat for domain
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sheaf
parent
5f98165276
commit
ed8720555f
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@ -61,18 +61,18 @@ type ParamsCt nbParams
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= ( Show ( ℝ nbParams )
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, HasChainRule Double 2 ( ℝ nbParams )
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, HasChainRule 𝕀 3 ( 𝕀ℝ nbParams )
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, Applicative ( D 2 ( ℝ nbParams ) )
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, Applicative ( D 3 ( ℝ nbParams ) )
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, Traversable ( D 2 ( ℝ nbParams ) )
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, Traversable ( D 3 ( ℝ nbParams ) )
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, Applicative ( D 2 nbParams )
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, Applicative ( D 3 nbParams )
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, Traversable ( D 2 nbParams )
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, Traversable ( D 3 nbParams )
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, Representable Double ( ℝ nbParams )
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, Representable 𝕀 ( 𝕀ℝ nbParams )
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, Module Double ( T ( ℝ nbParams ) )
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, Module 𝕀 ( T ( 𝕀ℝ nbParams ) )
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, Module ( D 2 ( ℝ nbParams ) Double ) ( D 2 ( ℝ nbParams ) ( ℝ 2 ) )
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, Module ( D 3 ( ℝ nbParams ) 𝕀 ) ( D 3 ( ℝ nbParams ) ( 𝕀ℝ 2 ) )
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, Transcendental ( D 2 ( ℝ nbParams ) Double )
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, Transcendental ( D 3 ( ℝ nbParams ) 𝕀 )
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, Module ( D 2 nbParams Double ) ( D 2 nbParams ( ℝ 2 ) )
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, Module ( D 3 nbParams 𝕀 ) ( D 3 nbParams ( 𝕀ℝ 2 ) )
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, Transcendental ( D 2 nbParams Double )
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, Transcendental ( D 3 nbParams 𝕀 )
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)
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newtype Params nbParams = Params { getParams :: ( ℝ nbParams ) }
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@ -75,10 +75,6 @@ data PointData params
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outlineFunction
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:: forall {t} (i :: t) brushParams
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. ( Show brushParams
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, D 1 ( I i 2 ) ~ D 1 ( ℝ 2 )
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, D 2 ( I i 2 ) ~ D 2 ( ℝ 2 )
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, D 3 ( I i 1 ) ~ D 3 ( ℝ 1 )
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, D 3 ( I i 2 ) ~ D 3 ( ℝ 2 )
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, HasType ( ℝ 2 ) ( PointData brushParams )
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, Cross ( I i Double ) ( T ( I i 2 ) )
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, Module ( I i Double ) ( T ( I i brushParams ) )
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@ -90,7 +86,6 @@ outlineFunction
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, HasChainRule ( I i Double ) 3 ( I i 1 )
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, HasChainRule ( I i Double ) 3 ( I i brushParams )
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, Traversable ( D 3 brushParams )
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, Traversable ( D 3 ( I i brushParams ) )
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, HasBézier 3 i
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)
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=> ( I i Double -> I i 1 )
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@ -263,9 +258,9 @@ instance HasBézier 3 AI where
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κ = 0.5519150244935105707435627227925
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circleSpline :: forall {t} (i :: t) k u v
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. Applicative ( D k ( I i u ) )
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=> ( Double -> Double -> D k ( I i u ) ( I i v ) )
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-> D k ( I i u ) ( Spline 'Closed () ( I i v ) )
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. Applicative ( D k ( Dim u ) )
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=> ( Double -> Double -> D k ( Dim u ) ( I i v ) )
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-> D k ( Dim u ) ( Spline 'Closed () ( I i v ) )
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circleSpline p = sequenceA $
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Spline { splineStart = p 1 0
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, splineCurves = ClosedCurves crvs lastCrv }
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@ -64,14 +64,15 @@ data Brush nbBrushParams
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-- Some convenience type synonyms for brush types... a bit horrible
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type ParamsICt :: Nat -> k -> Nat -> Constraint
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type ParamsICt k i rec =
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type ParamsICt k i nbParams =
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( Module
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( D k ( I i rec ) ( I i Double ) )
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( D k ( I i rec ) ( I i 2 ) )
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( D k nbParams ( I i Double ) )
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( D k nbParams ( I i 2 ) )
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, Module ( I i Double ) ( T ( I i Double ) )
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, HasChainRule ( I i Double ) k ( I i rec )
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, Representable ( I i Double ) ( I i rec )
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, Applicative ( D k ( I i rec ) )
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, HasChainRule ( I i Double ) k ( I i nbParams )
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, Representable ( I i Double ) ( I i nbParams )
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, Applicative ( D k nbParams )
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, Dim ( I i nbParams ) ~ nbParams
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)
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{-# INLINEABLE circleBrush #-}
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@ -137,19 +138,19 @@ circleBrushFn :: forall {t} (i :: t) k nbParams
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-> C k ( I i nbParams ) ( Spline 'Closed () ( I i 2 ) )
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circleBrushFn _ mkI1 mkI2 =
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D \ params ->
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let r :: D k ( I i nbParams ) ( I i Double )
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let r :: D k nbParams ( I i Double )
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r = runD ( var @_ @k $ Fin 1 ) params
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mkPt :: Double -> Double -> D k ( I i nbParams ) ( I i 2 )
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mkPt :: Double -> Double -> D k nbParams ( I i 2 )
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mkPt x y
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= ( r `scaledBy` x ) *^ e_x
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^+^ ( r `scaledBy` y ) *^ e_y
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in circleSpline mkPt
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where
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e_x, e_y :: D k ( I i nbParams ) ( I i 2 )
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e_x, e_y :: D k nbParams ( I i 2 )
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e_x = pure $ mkI2 $ ℝ2 1 0
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e_y = pure $ mkI2 $ ℝ2 0 1
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scaledBy :: D k ( I i nbParams ) ( I i Double ) -> Double -> D k ( I i nbParams ) ( I i Double )
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scaledBy :: D k nbParams ( I i Double ) -> Double -> D k nbParams ( I i Double )
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scaledBy d x = fmap ( mkI1 x * ) d
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{-# INLINEABLE circleBrushFn #-}
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@ -160,25 +161,24 @@ ellipseBrushFn :: forall {t} (i :: t) k nbParams
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=> Proxy# i
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-> ( Double -> I i Double )
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-> ( I ℝ 2 -> I i 2 )
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-> C k ( I i nbParams ) ( Spline 'Closed () ( I i 2 ) )
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ellipseBrushFn _ mkI1 mkI2 =
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D \ params ->
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let a, b :: D k ( I i nbParams ) ( I i Double )
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let a, b :: D k nbParams ( I i Double )
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a = runD ( var @_ @k $ Fin 1 ) params
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b = runD ( var @_ @k $ Fin 2 ) params
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mkPt :: Double -> Double -> D k ( I i nbParams ) ( I i 2 )
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mkPt :: Double -> Double -> D k nbParams ( I i 2 )
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mkPt x y
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= let !x' = a `scaledBy` x
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!y' = b `scaledBy` y
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in x' *^ e_x ^+^ y' *^ e_y
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in circleSpline mkPt
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where
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e_x, e_y :: D k ( I i nbParams ) ( I i 2 )
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e_x, e_y :: D k nbParams ( I i 2 )
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e_x = pure $ mkI2 $ ℝ2 1 0
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e_y = pure $ mkI2 $ ℝ2 0 1
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scaledBy :: D k ( I i nbParams ) ( I i Double ) -> Double -> D k ( I i nbParams ) ( I i Double )
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scaledBy :: D k nbParams ( I i Double ) -> Double -> D k nbParams ( I i Double )
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scaledBy d x = fmap ( mkI1 x * ) d
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{-# INLINEABLE ellipseBrushFn #-}
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@ -212,10 +212,10 @@ tearDropBrushFn :: forall {t} (i :: t) k nbParams
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-> C k ( I i nbParams ) ( Spline 'Closed () ( I i 2 ) )
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tearDropBrushFn _ mkI1 mkI2 =
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D \ params ->
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let w, h :: D k ( I i nbParams ) ( I i Double )
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let w, h :: D k nbParams ( I i Double )
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w = runD ( var @_ @k ( Fin 1 ) ) params
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h = runD ( var @_ @k ( Fin 2 ) ) params
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mkPt :: Double -> Double -> D k ( I i nbParams ) ( I i 2 )
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mkPt :: Double -> Double -> D k nbParams ( I i 2 )
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mkPt x y
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-- 1. translate the teardrop so that the centre of mass is at the origin
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-- 2. scale the teardrop so that it has the requested width/height
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@ -232,10 +232,10 @@ tearDropBrushFn _ mkI1 mkI2 =
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( mkPt -0.5 sqrt3_over_2 )
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BackToStart () }
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where
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e_x, e_y :: D k ( I i nbParams ) ( I i 2 )
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e_x, e_y :: D k nbParams ( I i 2 )
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e_x = pure $ mkI2 $ ℝ2 1 0
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e_y = pure $ mkI2 $ ℝ2 0 1
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scaledBy :: D k ( I i nbParams ) ( I i Double ) -> Double -> D k ( I i nbParams ) ( I i Double )
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scaledBy :: D k nbParams ( I i Double ) -> Double -> D k nbParams ( I i Double )
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scaledBy d x = fmap ( mkI1 x * ) d
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{-# INLINEABLE tearDropBrushFn #-}
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@ -1,4 +1,5 @@
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{-# LANGUAGE AllowAmbiguousTypes #-}
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{-# LANGUAGE PolyKinds #-}
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{-# LANGUAGE RebindableSyntax #-}
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{-# LANGUAGE ScopedTypeVariables #-}
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{-# LANGUAGE TemplateHaskell #-}
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@ -7,7 +8,7 @@
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{-# OPTIONS_GHC -Wno-orphans -O2 #-}
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module Math.Algebra.Dual
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( C(..), D
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( C(..), D, Dim
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, HasChainRule(..), chainRule
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, uncurryD2, uncurryD3
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, linear, fun, var
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@ -45,8 +46,12 @@ import Math.Ring
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-- | @C n u v@ is the space of @C^k@-differentiable maps from @u@ to @v@.
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type C :: Nat -> Type -> Type -> Type
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newtype C k u v = D { runD :: u -> D k u v }
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deriving stock instance Functor ( D k u ) => Functor ( C k u )
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newtype C k u v = D { runD :: u -> D k ( Dim u ) v }
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deriving stock instance Functor ( D k ( Dim u ) ) => Functor ( C k u )
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type Dim :: k -> Nat
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type family Dim u
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type instance Dim ( ℝ n ) = n
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-- | @D k u v@ is the space of @k@-th order germs of functions from @u@ to @v@,
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-- represented by the algebra:
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@ -54,35 +59,35 @@ deriving stock instance Functor ( D k u ) => Functor ( C k u )
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-- \[ \mathbb{Z}[x_1, \ldots, x_n]/(x_1, \ldots, x_n)^{k+1} \otimes_\mathbb{Z} v \]
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--
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-- when @u@ is of dimension @n@.
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type D :: Nat -> Type -> Type -> Type
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type D :: Nat -> Nat -> Type -> Type
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type family D k u
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type instance D k ( ℝ 0 ) = D𝔸0
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type instance D 0 ( ℝ 1 ) = D𝔸0
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type instance D 0 ( ℝ 2 ) = D𝔸0
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type instance D 0 ( ℝ 3 ) = D𝔸0
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type instance D 0 ( ℝ 4 ) = D𝔸0
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type instance D k 0 = D𝔸0
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type instance D 0 1 = D𝔸0
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type instance D 0 2 = D𝔸0
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type instance D 0 3 = D𝔸0
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type instance D 0 4 = D𝔸0
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type instance D 1 ( ℝ 1 ) = D1𝔸1
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type instance D 1 ( ℝ 2 ) = D1𝔸2
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type instance D 1 ( ℝ 3 ) = D1𝔸3
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type instance D 1 ( ℝ 4 ) = D1𝔸4
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type instance D 1 1 = D1𝔸1
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type instance D 1 2 = D1𝔸2
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type instance D 1 3 = D1𝔸3
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type instance D 1 4 = D1𝔸4
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type instance D 2 ( ℝ 1 ) = D2𝔸1
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type instance D 2 ( ℝ 2 ) = D2𝔸2
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type instance D 2 ( ℝ 3 ) = D2𝔸3
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type instance D 2 ( ℝ 4 ) = D2𝔸4
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type instance D 2 1 = D2𝔸1
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type instance D 2 2 = D2𝔸2
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type instance D 2 3 = D2𝔸3
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type instance D 2 4 = D2𝔸4
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type instance D 3 ( ℝ 1 ) = D3𝔸1
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type instance D 3 ( ℝ 2 ) = D3𝔸2
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type instance D 3 ( ℝ 3 ) = D3𝔸3
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type instance D 3 ( ℝ 4 ) = D3𝔸4
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type instance D 3 1 = D3𝔸1
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type instance D 3 2 = D3𝔸2
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type instance D 3 3 = D3𝔸3
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type instance D 3 4 = D3𝔸4
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--------------------------------------------------------------------------------
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-- Weird instance needed in just one place;
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-- see use of chain in 'Math.Bezier.Stroke.brushStrokeData'.
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instance ( Applicative ( D k u ), Module r ( T v ) )
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instance ( Applicative ( D k ( Dim u ) ), Module r ( T v ) )
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=> Module r ( T ( C k u v ) ) where
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origin = T $ D \ _ -> pure $ coerce $ origin @r @( T v )
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T ( D f ) ^+^ T ( D g ) = T $ D \ t -> liftA2 ( coerce $ (^+^) @r @( T v ) ) ( f t ) ( g t )
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@ -98,10 +103,10 @@ instance ( Applicative ( D k u ), Module r ( T v ) )
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-- with @D k v w@ in the middle, for any @r@-module @w@.
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class HasChainRule r k v where
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chain :: Module r ( T w )
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=> D k ( ℝ 1 ) v -> D k v w -> D k ( ℝ 1 ) w
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konst :: AbelianGroup w => w -> D k v w
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value :: D k v w -> w
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linearD :: Module r ( T w ) => ( v -> w ) -> v -> D k v w
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=> D k 1 v -> D k ( Dim v ) w -> D k 1 w
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konst :: AbelianGroup w => w -> D k ( Dim v ) w
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value :: D k ( Dim v ) w -> w
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linearD :: Module r ( T w ) => ( v -> w ) -> v -> D k ( Dim v ) w
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linear :: forall k r v w
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. ( HasChainRule r k v, Module r ( T w ) )
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@ -110,7 +115,7 @@ linear f = D \ x -> linearD @r @k @v @w f x
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chainRule :: forall r k u v w
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. ( HasChainRule r k v, Module r ( T w )
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, D k u ~ D k ( ℝ 1 ), HasChainRule r k u
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, Dim u ~ 1, HasChainRule r k u
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)
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=> C k u v -> C k v w -> C k u w
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chainRule ( D df ) ( D dg ) =
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@ -120,8 +125,8 @@ chainRule ( D df ) ( D dg ) =
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chain @r @k @v df_x ( dg $ value @r @k @u df_x )
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uncurryD2 :: D 2 a ~ D 2 ( ℝ 1 )
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=> D 2 ( ℝ 1 ) ( C 2 a b ) -> a -> D 2 ( ℝ 2 ) b
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uncurryD2 :: Dim a ~ 1
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=> D 2 1 ( C 2 a b ) -> a -> D 2 2 b
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uncurryD2 ( D21 ( D b_t0 ) ( T ( D dbdt_t0 ) ) ( T ( D d2bdt2_t0 ) ) ) s0 =
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let D21 b_t0s0 dbds_t0s0 d2bds2_t0s0 = b_t0 s0
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D21 dbdt_t0s0 d2bdtds_t0s0 _ = dbdt_t0 s0
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@ -131,8 +136,8 @@ uncurryD2 ( D21 ( D b_t0 ) ( T ( D dbdt_t0 ) ) ( T ( D d2bdt2_t0 ) ) ) s0 =
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( T dbdt_t0s0 ) dbds_t0s0
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( T d2bdt2_t0s0 ) d2bdtds_t0s0 d2bds2_t0s0
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uncurryD3 :: D 3 a ~ D 3 ( ℝ 1 )
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=> D 3 ( ℝ 1 ) ( C 3 a b ) -> a -> D 3 ( ℝ 2 ) b
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uncurryD3 :: Dim a ~ 1
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=> D 3 1 ( C 3 a b ) -> a -> D 3 2 b
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uncurryD3 ( D31 ( D b_t0 ) ( T ( D dbdt_t0 ) ) ( T ( D d2bdt2_t0 ) ) ( T ( D d3bdt3_t0 ) ) ) s0 =
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let D31 b_t0s0 dbds_t0s0 d2bds2_t0s0 d3bds3_t0s0 = b_t0 s0
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D31 dbdt_t0s0 d2bdtds_t0s0 d3bdtds2_t0s0 _ = dbdt_t0 s0
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@ -159,6 +164,7 @@ var i = D $ linearD @r @k @v ( `index` i )
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--------------------------------------------------------------------------------
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-- | Newtype for the module instance @Module r v => Module ( dr r ) ( dr v )@.
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type ApAp :: Type -> ( Type -> Type ) -> Type -> Type
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newtype ApAp r dr v = ApAp { unApAp :: dr v }
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instance ( Ring ( dr r ), Module r ( T v ), Applicative dr )
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@ -199,9 +205,9 @@ deriving via ApAp r D3𝔸4 v
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--------------------------------------------------------------------------------
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-- AbelianGroup instances
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newtype ApAp2 k u r = ApAp2 { unApAp2 :: D k u r }
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newtype ApAp2 k u r = ApAp2 { unApAp2 :: D k ( Dim u ) r }
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instance ( Applicative ( D k u )
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instance ( Applicative ( D k ( Dim u ) )
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, AbelianGroup r
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, HasChainRule Double k u
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) => AbelianGroup ( ApAp2 k u r ) where
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@ -62,7 +62,7 @@ import GHC.STRef
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import GHC.Generics
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( Generic, Generic1, Generically(..) )
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import GHC.TypeNats
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( Nat, type (-) )
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( Nat )
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-- acts
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import Data.Act
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@ -213,7 +213,7 @@ data Cusp
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= Cusp
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{ cuspParameters :: !( ℝ 2 )
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-- ^ @(t,s)@ parameter values of the cusp
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, cuspPathCoords :: !( D 2 ( ℝ 1 ) ( ℝ 2 ) )
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, cuspPathCoords :: !( D 2 1 ( ℝ 2 ) )
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-- ^ path point coordinates and tangent
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, cuspStrokeCoords :: !( ℝ 2 )
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-- ^ brush stroke point coordinates
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@ -243,10 +243,11 @@ computeStrokeOutline ::
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, HasChainRule 𝕀 3 ( 𝕀ℝ nbUsedParams )
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, HasChainRule Double 2 ( ℝ nbBrushParams )
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, HasChainRule 𝕀 3 ( 𝕀ℝ nbBrushParams )
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, Traversable ( D 2 ( ℝ nbBrushParams ) )
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, Traversable ( D 2 nbBrushParams )
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, Traversable ( D 3 nbBrushParams )
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, Representable Double ( ℝ nbUsedParams )
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, Representable 𝕀 ( 𝕀ℝ nbUsedParams )
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, Module 𝕀 (T (𝕀ℝ nbUsedParams))
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, Module 𝕀 (T ( 𝕀ℝ nbUsedParams ) )
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-- Debugging.
|
||||
, Show ptData, Show ( ℝ nbBrushParams )
|
||||
|
|
@ -526,8 +527,9 @@ outlineFunction
|
|||
, HasChainRule 𝕀 3 ( 𝕀ℝ nbUsedParams )
|
||||
, HasChainRule Double 2 ( ℝ nbBrushParams )
|
||||
, HasChainRule 𝕀 3 ( 𝕀ℝ nbBrushParams )
|
||||
, Traversable ( D 2 ( ℝ nbBrushParams ) )
|
||||
, Module 𝕀 (T (𝕀ℝ nbUsedParams))
|
||||
, Traversable ( D 2 nbBrushParams )
|
||||
, Traversable ( D 3 nbBrushParams )
|
||||
, Module 𝕀 ( T ( 𝕀ℝ nbUsedParams ) )
|
||||
|
||||
-- Computing AABBs
|
||||
, Representable Double ( ℝ nbUsedParams )
|
||||
|
|
@ -624,7 +626,6 @@ pathAndUsedParams :: forall k i (nbUsedParams :: Nat) arr crvData ptData
|
|||
. ( HasType ( ℝ 2 ) ptData
|
||||
, HasBézier k i
|
||||
, arr ~ C k
|
||||
, D k ( I i 1 ) ~ D k ( ℝ 1 )
|
||||
, Module ( I i Double ) ( T ( I i 2 ) )
|
||||
, Torsor ( T ( I i 2 ) ) ( I i 2 )
|
||||
, Module ( I i Double ) ( T ( I i nbUsedParams ) )
|
||||
|
|
@ -955,7 +956,6 @@ withTangent tgt_wanted spline@( Spline { splineStart } )
|
|||
|
||||
splineCurveFns :: forall k i
|
||||
. ( HasBézier k i
|
||||
, D k ( I i 1 ) ~ D k ( ℝ 1 )
|
||||
, Module ( I i Double ) ( T ( I i 2 ) )
|
||||
, Torsor ( T ( I i 2 ) ) ( I i 2 ) )
|
||||
=> ( I i 1 -> I i Double )
|
||||
|
|
@ -992,13 +992,11 @@ brushStrokeData :: forall {kd} (k :: Nat) (nbBrushParams :: Nat) (i :: kd) arr
|
|||
, HasBézier k i, HasEnvelopeEquation k
|
||||
, Differentiable k i nbBrushParams
|
||||
, HasChainRule ( I i Double ) k ( I i 1 )
|
||||
, Applicative ( D k ( ℝ 1 ) )
|
||||
, Applicative ( D k 1 )
|
||||
, Dim ( I i 1 ) ~ 1
|
||||
, Dim ( I i nbBrushParams ) ~ nbBrushParams
|
||||
, Traversable ( D k nbBrushParams )
|
||||
|
||||
, D ( k - 2 ) ( I i 2 ) ~ D ( k - 2 ) ( ℝ 2 )
|
||||
, D ( k - 1 ) ( I i 2 ) ~ D ( k - 1 ) ( ℝ 2 )
|
||||
, D k ( I i 1 ) ~ D k ( ℝ 1 )
|
||||
, D k ( I i 1 ) ~ D k ( ℝ 1 )
|
||||
, D k ( I i 2 ) ~ D k ( ℝ 2 )
|
||||
, Transcendental ( I i Double )
|
||||
, Module ( I i Double ) ( T ( I i 1 ) )
|
||||
, Cross ( I i Double ) ( T ( I i 2 ) )
|
||||
|
|
@ -1020,24 +1018,24 @@ brushStrokeData :: forall {kd} (k :: Nat) (nbBrushParams :: Nat) (i :: kd) arr
|
|||
brushStrokeData co1 co2 path params brush mbBrushRotation =
|
||||
\ t ->
|
||||
let
|
||||
dpath_t :: D k ( I i 1 ) ( I i 2 )
|
||||
dpath_t :: D k 1 ( I i 2 )
|
||||
!dpath_t = runD path t
|
||||
dparams_t :: D k ( I i 1 ) ( I i nbBrushParams )
|
||||
dparams_t :: D k 1 ( I i nbBrushParams )
|
||||
!dparams_t = runD params t
|
||||
dbrush_params :: D k ( I i nbBrushParams ) ( Spline Closed () ( I i 2 ) )
|
||||
dbrush_params :: D k nbBrushParams ( Spline Closed () ( I i 2 ) )
|
||||
!dbrush_params = runD brush $ value @( I i Double ) @k @( I i 1 ) dparams_t
|
||||
splines :: Seq ( D k ( I i nbBrushParams ) ( I i 1 `arr` I i 2 ) )
|
||||
splines :: Seq ( D k nbBrushParams ( I i 1 `arr` I i 2 ) )
|
||||
!splines = getZipSeq $ traverse ( ZipSeq . splineCurveFns @k @i co2 ) dbrush_params
|
||||
dbrushes_t :: Seq ( I i 1 -> D k ( I i 2 ) ( I i 2 ) )
|
||||
dbrushes_t :: Seq ( I i 1 -> D k 2 ( I i 2 ) )
|
||||
!dbrushes_t = force $ fmap ( uncurryD @k . chain @( I i Double ) @k dparams_t ) splines
|
||||
-- This is the crucial use of the chain rule.
|
||||
|
||||
in fmap ( mkStrokeDatum dpath_t dparams_t ) dbrushes_t
|
||||
where
|
||||
|
||||
mkStrokeDatum :: D k ( I i 1 ) ( I i 2 )
|
||||
-> D k ( I i 1 ) ( I i nbBrushParams )
|
||||
-> ( I i 1 -> D k ( I i 2 ) ( I i 2 ) )
|
||||
mkStrokeDatum :: D k 1 ( I i 2 )
|
||||
-> D k 1 ( I i nbBrushParams )
|
||||
-> ( I i 1 -> D k 2 ( I i 2 ) )
|
||||
-> ( I i 1 -> StrokeDatum k i )
|
||||
mkStrokeDatum dpath_t dparams_t dbrush_t s =
|
||||
let dbrush_t_s = dbrush_t s
|
||||
|
|
|
|||
|
|
@ -45,11 +45,11 @@ type StrokeDatum :: Nat -> k -> Type
|
|||
data StrokeDatum k i
|
||||
= StrokeDatum
|
||||
{ -- | Path \( p(t) \).
|
||||
dpath :: D k ( I i 1 ) ( I i 2 )
|
||||
dpath :: D k 1 ( I i 2 )
|
||||
-- | Brush shape \( b(t, s) \).
|
||||
, dbrush :: D k ( I i 2 ) ( I i 2 )
|
||||
, dbrush :: D k 2 ( I i 2 )
|
||||
-- | (Optional) rotation angle \( \theta(t) \).
|
||||
, mbRotation :: Maybe ( D k ( I i 1 ) ( I i Double ) )
|
||||
, mbRotation :: Maybe ( D k 1 ( I i Double ) )
|
||||
|
||||
-- Everything below is computed in terms of the first three fields.
|
||||
|
||||
|
|
@ -58,19 +58,19 @@ data StrokeDatum k i
|
|||
|
||||
-- | \( u(t,s) = R(-\theta(t)) \frac{\partial c}{\partial t} \),
|
||||
-- \( v(t,s) = R(-\theta(t)) \frac{\partial c}{\partial s} \)
|
||||
, du, dv :: D ( k - 1 ) ( I i 2 ) ( I i 2 )
|
||||
, du, dv :: D ( k - 1 ) 2 ( I i 2 )
|
||||
|
||||
-- | Envelope function
|
||||
--
|
||||
-- \[ E(t_0,s_0) = \left ( \frac{\partial c}{\partial t} \times \frac{\partial c}{\partial s} \right )_{(t_0,s_0)}. \]
|
||||
, ee :: D ( k - 1 ) ( I i 2 ) ( I i 1 )
|
||||
, ee :: D ( k - 1 ) 2 ( I i 1 )
|
||||
|
||||
-- \[ \frac{\partial E}{\partial s} \frac{\mathrm{d} c}{\mathrm{d} t}, \]
|
||||
--
|
||||
-- where \( \frac{\mathrm{d} c}{\mathrm{d} t} \)
|
||||
--
|
||||
-- denotes a total derivative.
|
||||
, 𝛿E𝛿sdcdt :: D ( k - 2 ) ( I i 2 ) ( T ( I i 2 ) )
|
||||
, 𝛿E𝛿sdcdt :: D ( k - 2 ) 2 ( T ( I i 2 ) )
|
||||
}
|
||||
|
||||
deriving stock instance Show ( StrokeDatum 2 ℝ )
|
||||
|
|
@ -85,7 +85,6 @@ class HasBézier k i where
|
|||
line :: forall ( n :: Nat )
|
||||
. ( Module Double ( T ( ℝ n ) ), Torsor ( T ( ℝ n ) ) ( ℝ n )
|
||||
, Module ( I i Double ) ( T ( I i n ) ), Torsor ( T ( I i n ) ) ( I i n )
|
||||
, D k ( I i 1 ) ~ D k ( ℝ 1 )
|
||||
)
|
||||
=> ( I i 1 -> I i Double )
|
||||
-> Segment ( I i n ) -> C k ( I i 1 ) ( I i n )
|
||||
|
|
@ -96,7 +95,6 @@ class HasBézier k i where
|
|||
, Module ( I i Double ) ( T ( I i n ) ), Torsor ( T ( I i n ) ) ( I i n )
|
||||
, Representable Double ( ℝ n )
|
||||
, Representable 𝕀 ( 𝕀ℝ n )
|
||||
, D k ( I i 1 ) ~ D k ( ℝ 1 )
|
||||
)
|
||||
=> ( I i 1 -> I i Double )
|
||||
-> Quadratic.Bezier ( I i n ) -> C k ( I i 1 ) ( I i n )
|
||||
|
|
@ -107,7 +105,6 @@ class HasBézier k i where
|
|||
, Module ( I i Double ) ( T ( I i n ) ), Torsor ( T ( I i n ) ) ( I i n )
|
||||
, Representable Double ( ℝ n )
|
||||
, Representable 𝕀 ( 𝕀ℝ n )
|
||||
, D k ( I i 1 ) ~ D k ( ℝ 1 )
|
||||
)
|
||||
=> ( I i 1 -> I i Double )
|
||||
-> Cubic.Bezier ( I i n ) -> C k ( I i 1 ) ( I i n )
|
||||
|
|
@ -115,8 +112,8 @@ class HasBézier k i where
|
|||
type HasEnvelopeEquation :: Nat -> Constraint
|
||||
class HasEnvelopeEquation k where
|
||||
|
||||
uncurryD :: D k a ~ D k ( ℝ 1 )
|
||||
=> D k ( ℝ 1 ) ( C k a b ) -> a -> D k ( ℝ 2 ) b
|
||||
uncurryD :: Dim a ~ 1
|
||||
=> D k 1 ( C k a b ) -> a -> D k 2 b
|
||||
|
||||
-- | The envelope function
|
||||
--
|
||||
|
|
@ -130,11 +127,7 @@ class HasEnvelopeEquation k where
|
|||
--
|
||||
-- denotes a total derivative.
|
||||
envelopeEquation :: forall i
|
||||
. ( D ( k - 2 ) ( I i 2 ) ~ D ( k - 2 ) ( ℝ 2 )
|
||||
, D ( k - 1 ) ( I i 2 ) ~ D ( k - 1 ) ( ℝ 2 )
|
||||
, D k ( I i 2 ) ~ D k ( ℝ 2 )
|
||||
, D k ( I i 1 ) ~ D k ( ℝ 1 )
|
||||
, Module ( I i Double ) ( T ( I i 1 ) )
|
||||
. ( Module ( I i Double ) ( T ( I i 1 ) )
|
||||
, Cross ( I i Double ) ( T ( I i 2 ) )
|
||||
, Transcendental ( I i Double )
|
||||
, Representable ( I i Double ) ( I i 2 )
|
||||
|
|
@ -142,9 +135,9 @@ class HasEnvelopeEquation k where
|
|||
)
|
||||
=> Proxy i
|
||||
-> ( I i Double -> I i 1 )
|
||||
-> D k ( I i 1 ) ( I i 2 )
|
||||
-> D k ( I i 2 ) ( I i 2 )
|
||||
-> Maybe ( D k ( I i 1 ) ( I i Double ) )
|
||||
-> D k 1 ( I i 2 )
|
||||
-> D k 2 ( I i 2 )
|
||||
-> Maybe ( D k 1 ( I i Double ) )
|
||||
-> StrokeDatum k i
|
||||
|
||||
instance HasBézier 2 ℝ where
|
||||
|
|
|
|||
|
|
@ -37,12 +37,12 @@ type Differentiable :: Nat -> k -> Nat -> Constraint
|
|||
class
|
||||
( Module ( I i Double ) ( T ( I i u ) )
|
||||
, HasChainRule ( I i Double ) k ( I i u )
|
||||
, Traversable ( D k ( I i u ) )
|
||||
, Traversable ( D k u )
|
||||
) => Differentiable k i u
|
||||
instance
|
||||
( Module ( I i Double ) ( T ( I i u ) )
|
||||
, HasChainRule ( I i Double ) k ( I i u )
|
||||
, Traversable ( D k ( I i u ) )
|
||||
, Traversable ( D k u )
|
||||
) => Differentiable k i u
|
||||
|
||||
type DiffInterp :: Nat -> k -> Nat -> Constraint
|
||||
|
|
@ -50,10 +50,10 @@ class
|
|||
( Differentiable k i u
|
||||
, Interpolatable ( I i Double ) ( I i u )
|
||||
, Module ( I i Double ) ( T ( I i Double ) )
|
||||
, Module ( D k ( I i u ) ( I i Double ) )
|
||||
( D k ( I i u ) ( I i 2 ) )
|
||||
, Transcendental ( D k ( I i u ) ( I i Double ) )
|
||||
, Applicative ( D k ( I i u ) )
|
||||
, Module ( D k u ( I i Double ) )
|
||||
( D k u ( I i 2 ) )
|
||||
, Transcendental ( D k u ( I i Double ) )
|
||||
, Applicative ( D k u )
|
||||
, Representable ( I i Double ) ( I i u )
|
||||
, RepDim ( I i u ) ~ RepDim u
|
||||
) => DiffInterp k i u
|
||||
|
|
@ -61,10 +61,10 @@ instance
|
|||
( Differentiable k i u
|
||||
, Interpolatable ( I i Double ) ( I i u )
|
||||
, Module ( I i Double ) ( T ( I i Double ) )
|
||||
, Module ( D k ( I i u ) ( I i Double ) )
|
||||
( D k ( I i u ) ( I i 2 ) )
|
||||
, Transcendental ( D k ( I i u ) ( I i Double ) )
|
||||
, Applicative ( D k ( I i u ) )
|
||||
, Module ( D k u ( I i Double ) )
|
||||
( D k u ( I i 2 ) )
|
||||
, Transcendental ( D k u ( I i Double ) )
|
||||
, Applicative ( D k u )
|
||||
, Representable ( I i Double ) ( I i u )
|
||||
, RepDim ( I i u ) ~ RepDim u
|
||||
) => DiffInterp k i u
|
||||
|
|
|
|||
|
|
@ -58,8 +58,7 @@ import Math.Ring
|
|||
--------------------------------------------------------------------------------
|
||||
-- Interval arithmetic.
|
||||
|
||||
type instance D k 𝕀 = D k Double
|
||||
type instance D k ( 𝕀ℝ n ) = D k ( ℝ n )
|
||||
type instance Dim ( 𝕀ℝ n ) = n
|
||||
|
||||
-- | Turn a non-decreasing function into a function on intervals.
|
||||
nonDecreasing :: forall n m
|
||||
|
|
|
|||
|
|
@ -188,7 +188,7 @@ isolateRootsIn
|
|||
. BoxCt n d
|
||||
=> RootIsolationOptions n d
|
||||
-- ^ configuration (which algorithms to use, and with what parameters)
|
||||
-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
|
||||
-> ( 𝕀ℝ n -> D 1 n ( 𝕀ℝ d ) )
|
||||
-- ^ equations to solve
|
||||
-> Box n
|
||||
-- ^ initial search domain
|
||||
|
|
@ -271,7 +271,7 @@ doStrategy
|
|||
. RootIsolationAlgorithmWithOptions n d
|
||||
-> [ ( RootIsolationStep, Box n ) ]
|
||||
-> BoxHistory n
|
||||
-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
|
||||
-> ( 𝕀ℝ n -> D 1 n ( 𝕀ℝ d ) )
|
||||
-> Box n
|
||||
-> Writer ( DoneBoxes n )
|
||||
( RootIsolationStep, [ Box n ] )
|
||||
|
|
|
|||
|
|
@ -74,7 +74,7 @@ instance BoxCt n d => RootIsolationAlgorithm Bisection n d where
|
|||
:: RootIsolationAlgorithmOptions Bisection 2 3
|
||||
-> [ ( RootIsolationStep, Box 2 ) ]
|
||||
-> BoxHistory 2
|
||||
-> ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
|
||||
-> ( 𝕀ℝ 2 -> D 1 2 ( 𝕀ℝ 3 ) )
|
||||
-> Box 2
|
||||
-> Writer ( DoneBoxes 2 ) ( StepDescription Bisection, [ Box 2 ] ) #-}
|
||||
-- NB: including this to be safe. The specialiser seems to sometimes
|
||||
|
|
@ -92,7 +92,7 @@ data BisectionOptions n d =
|
|||
--
|
||||
-- NB: only return 'False' if non-existence of solutions is guaranteed
|
||||
-- (otherwise, the root isolation algorithm might not be consistent).
|
||||
canHaveSols :: !( ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) ) -> Box n -> Bool )
|
||||
canHaveSols :: !( ( 𝕀ℝ n -> D 1 n ( 𝕀ℝ d ) ) -> Box n -> Bool )
|
||||
-- | Heuristic to choose which coordinate dimension to bisect.
|
||||
--
|
||||
-- It's only a fallback, as we prefer to bisect along coordinate dimensions
|
||||
|
|
@ -104,7 +104,7 @@ data BisectionOptions n d =
|
|||
type BisectionCoordPicker n d
|
||||
= [ ( RootIsolationStep, Box n ) ]
|
||||
-> BoxHistory n
|
||||
-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
|
||||
-> ( 𝕀ℝ n -> D 1 n ( 𝕀ℝ d ) )
|
||||
-> forall r. ( NE.NonEmpty ( Fin n, r ) -> ( r, String ) )
|
||||
|
||||
-- | Default options for the bisection method.
|
||||
|
|
|
|||
|
|
@ -98,9 +98,9 @@ type BoxCt n d =
|
|||
, Eq ( ℝ n )
|
||||
, Representable Double ( ℝ n )
|
||||
, Representable 𝕀 ( 𝕀ℝ n )
|
||||
, MonomialBasis ( D 1 ( ℝ n ) )
|
||||
, Deg ( D 1 ( ℝ n ) ) ~ 1
|
||||
, Vars ( D 1 ( ℝ n ) ) ~ n
|
||||
, MonomialBasis ( D 1 n )
|
||||
, Deg ( D 1 n ) ~ 1
|
||||
, Vars ( D 1 n ) ~ n
|
||||
, Module Double ( T ( ℝ n ) )
|
||||
, Module 𝕀 ( T ( 𝕀ℝ n ) )
|
||||
, NFData ( ℝ n )
|
||||
|
|
@ -181,7 +181,7 @@ class ( Typeable ty, Show ( StepDescription ty ), BoxCt n d )
|
|||
-- ^ history of the current round
|
||||
-> BoxHistory n
|
||||
-- ^ previous rounds history
|
||||
-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
|
||||
-> ( 𝕀ℝ n -> D 1 n ( 𝕀ℝ d ) )
|
||||
-- ^ equations
|
||||
-> Box n
|
||||
-- ^ box
|
||||
|
|
|
|||
|
|
@ -75,7 +75,7 @@ instance BoxCt n d => RootIsolationAlgorithm Box1 n d where
|
|||
:: RootIsolationAlgorithmOptions Box1 2 3
|
||||
-> [ ( RootIsolationStep, Box 2 ) ]
|
||||
-> BoxHistory 2
|
||||
-> ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
|
||||
-> ( 𝕀ℝ 2 -> D 1 2 ( 𝕀ℝ 3 ) )
|
||||
-> Box 2
|
||||
-> Writer ( DoneBoxes 2 ) ( StepDescription Box1, [ Box 2 ] ) #-}
|
||||
-- NB: including this to be safe. The specialiser seems to sometimes
|
||||
|
|
@ -93,7 +93,7 @@ instance BoxCt n d => RootIsolationAlgorithm Box2 n d where
|
|||
:: RootIsolationAlgorithmOptions Box2 2 3
|
||||
-> [ ( RootIsolationStep, Box 2 ) ]
|
||||
-> BoxHistory 2
|
||||
-> ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
|
||||
-> ( 𝕀ℝ 2 -> D 1 2 ( 𝕀ℝ 3 ) )
|
||||
-> Box 2
|
||||
-> Writer ( DoneBoxes 2 ) ( StepDescription Box2, [ Box 2 ] ) #-}
|
||||
-- NB: including this to be safe. The specialiser seems to sometimes
|
||||
|
|
@ -159,14 +159,14 @@ makeBox1Consistent
|
|||
:: ( KnownNat n
|
||||
, Representable Double ( ℝ n )
|
||||
, Representable 𝕀 ( 𝕀ℝ n )
|
||||
, MonomialBasis ( D 1 ( ℝ n ) )
|
||||
, Deg ( D 1 ( ℝ n ) ) ~ 1
|
||||
, Vars ( D 1 ( ℝ n ) ) ~ n
|
||||
, MonomialBasis ( D 1 n )
|
||||
, Deg ( D 1 n ) ~ 1
|
||||
, Vars ( D 1 n ) ~ n
|
||||
, Representable Double ( ℝ d )
|
||||
, Representable 𝕀 ( 𝕀ℝ d )
|
||||
)
|
||||
=> Box1Options n d
|
||||
-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
|
||||
-> ( 𝕀ℝ n -> D 1 n ( 𝕀ℝ d ) )
|
||||
-> 𝕀ℝ n -> [ 𝕀ℝ n ]
|
||||
makeBox1Consistent box1Options eqs x =
|
||||
( `State.evalState` False ) $
|
||||
|
|
@ -180,14 +180,14 @@ makeBox2Consistent
|
|||
. ( KnownNat n
|
||||
, Representable Double ( ℝ n )
|
||||
, Representable 𝕀 ( 𝕀ℝ n )
|
||||
, MonomialBasis ( D 1 ( ℝ n ) )
|
||||
, Deg ( D 1 ( ℝ n ) ) ~ 1
|
||||
, Vars ( D 1 ( ℝ n ) ) ~ n
|
||||
, MonomialBasis ( D 1 n )
|
||||
, Deg ( D 1 n ) ~ 1
|
||||
, Vars ( D 1 n ) ~ n
|
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, Representable Double ( ℝ d )
|
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, Representable 𝕀 ( 𝕀ℝ d )
|
||||
)
|
||||
=> Box2Options n d
|
||||
-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
|
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-> ( 𝕀ℝ n -> D 1 n ( 𝕀ℝ d ) )
|
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-> 𝕀ℝ n -> 𝕀ℝ n
|
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makeBox2Consistent (Box2Options box1Options ε_eq λMin) eqs x0
|
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= ( `State.evalState` False ) $ doLoop 0.25 x0
|
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|
|
@ -261,14 +261,14 @@ allNarrowingOperators
|
|||
. ( KnownNat n
|
||||
, Representable Double ( ℝ n )
|
||||
, Representable 𝕀 ( 𝕀ℝ n )
|
||||
, MonomialBasis ( D 1 ( ℝ n ) )
|
||||
, Deg ( D 1 ( ℝ n ) ) ~ 1
|
||||
, Vars ( D 1 ( ℝ n ) ) ~ n
|
||||
, MonomialBasis ( D 1 n )
|
||||
, Deg ( D 1 n ) ~ 1
|
||||
, Vars ( D 1 n ) ~ n
|
||||
, Representable Double ( ℝ d )
|
||||
, Representable 𝕀 ( 𝕀ℝ d )
|
||||
)
|
||||
=> Box1Options n d
|
||||
-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
|
||||
-> ( 𝕀ℝ n -> D 1 n ( 𝕀ℝ d ) )
|
||||
-> [ 𝕀ℝ n -> State Bool [ 𝕀ℝ n ] ]
|
||||
allNarrowingOperators ( Box1Options ε_eq ε_bis coordsToNarrow eqsToUse narrowingMethod ) eqs =
|
||||
[ \ cand ->
|
||||
|
|
|
|||
|
|
@ -63,7 +63,7 @@ instance BoxCt n d => RootIsolationAlgorithm Newton n d where
|
|||
:: RootIsolationAlgorithmOptions Newton 2 3
|
||||
-> [ ( RootIsolationStep, Box 2 ) ]
|
||||
-> BoxHistory 2
|
||||
-> ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
|
||||
-> ( 𝕀ℝ 2 -> D 1 2 ( 𝕀ℝ 3 ) )
|
||||
-> Box 2
|
||||
-> Writer ( DoneBoxes 2 ) ( StepDescription Newton, [ Box 2 ] ) #-}
|
||||
-- NB: including this to be safe. The specialiser seems to sometimes
|
||||
|
|
@ -100,7 +100,7 @@ intervalNewton
|
|||
:: forall n d
|
||||
. BoxCt n d
|
||||
=> NewtonOptions n d
|
||||
-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
|
||||
-> ( 𝕀ℝ n -> D 1 n ( 𝕀ℝ d ) )
|
||||
-- ^ equations
|
||||
-> 𝕀ℝ n
|
||||
-- ^ box
|
||||
|
|
|
|||
Loading…
Reference in a new issue