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add separate EnvelopeEquation module (forgot to add earlier)
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src/splines/Math/Bezier/Stroke/EnvelopeEquation.hs
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246
src/splines/Math/Bezier/Stroke/EnvelopeEquation.hs
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{-# LANGUAGE AllowAmbiguousTypes #-}
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{-# LANGUAGE RebindableSyntax #-}
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{-# LANGUAGE ScopedTypeVariables #-}
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module Math.Bezier.Stroke.EnvelopeEquation
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( StrokeDatum(..)
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, CurveOrder(..)
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) where
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-- base
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import Prelude hiding ( Num(..), (^) )
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import Data.Coerce
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( Coercible, coerce )
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import Data.Kind
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( Constraint )
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import GHC.TypeNats
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( Nat, type (-) )
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-- acts
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import Data.Act
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( Torsor ((-->)) )
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-- MetaBrush
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import Math.Algebra.Dual
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import qualified Math.Bezier.Cubic as Cubic
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import qualified Math.Bezier.Quadratic as Quadratic
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import Math.Differentiable
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( type ExtentOrder )
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import Math.Interval
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import Math.Linear
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import Math.Module
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( Module(..), Cross((×)), lerp )
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import Math.Ring
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--------------------------------------------------------------------------------
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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 i
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= StrokeDatum
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{ -- | Path \( p(t_0) \).
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dpath :: D ( ExtentOrder i ) ( I i ( ℝ 1 ) ) ( I i ( ℝ 2 ) )
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-- | Brush shape \( b(t_0, s_0) \).
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, dbrush :: D ( ExtentOrder i ) ( 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 \( c(t_0,s_0) = p(t_0) + b(t_0,s_0) \).
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, dstroke :: D ( ExtentOrder i ) ( I i ( ℝ 2 ) ) ( I i ( ℝ 2 ) )
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-- | Envelope function
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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 :: D ( ExtentOrder i - 1 ) ( I i ( ℝ 2 ) ) ( I i ( ℝ 1 ) )
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-- \[ \frac{\partial E}{\partial s} \frac{\mathrm{d} c}{\mathrm{d} t}, \]
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--
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-- where \( \frac{\mathrm{d} c}{\mathrm{d} t} \)
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--
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-- denotes a total derivative.
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, 𝛿E𝛿sdcdt :: D ( ExtentOrder i - 2 ) ( I i ( ℝ 2 ) ) ( T ( I i ( ℝ 2 ) ) )
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}
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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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type CurveOrder :: Nat -> Constraint
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class CurveOrder k where
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-- | Linear interpolation, as a differentiable function.
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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 k ( I i ( ℝ 1 ) ) ~ D k ( ℝ 1 )
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, Coercible ( I i ( ℝ 1 ) ) ( I i Double )
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)
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=> Segment b -> C k ( I i ( ℝ 1 ) ) b
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-- | A quadratic Bézier curve, as a differentiable function.
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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 k ( I i ( ℝ 1 ) ) ~ D k ( 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 -> C k ( I i ( ℝ 1 ) ) b
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-- | A cubic Bézier curve, as a differentiable function.
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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 k ( I i ( ℝ 1 ) ) ~ D k ( 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 -> C k ( I i ( ℝ 1 ) ) b
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uncurryD :: D k a ~ D k ( ℝ 1 )
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=> D k ( ℝ 1 ) ( C k a b ) -> a -> D k ( ℝ 2 ) b
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-- | A brush stroke, as described by the equation
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--
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-- \[ c(t,s) = p(t) + b(t,s) \]
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--
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-- where:
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--
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-- - \( p(t) \) is the path that the brush follows, and
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-- - \( b(t,s) \) is the brush shape, as it varies along the path.
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brushStroke :: Module r ( T v )
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=> D k ( ℝ 1 ) v -- ^ stroke path \( p(t) \)
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-> D k ( ℝ 2 ) v -- ^ brush \( b(t,s) \)
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-> D k ( ℝ 2 ) v
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-- | The envelope function
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--
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-- \[ E = \frac{\partial c}{\partial t} \times \frac{\partial c}{\partial s}, \]
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--
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-- together with the 2-vector
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--
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-- \[ \frac{\partial E}{\partial s} \frac{\mathrm{d} c}{\mathrm{d} t}, \]
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--
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-- where \( \frac{\mathrm{d} c}{\mathrm{d} t} \)
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--
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-- denotes a total derivative.
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envelopeEquation :: forall i
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. ( k ~ ExtentOrder i
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, D ( k - 2 ) ( I i ( ℝ 2 ) ) ~ D ( k - 2 ) ( ℝ 2 )
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, D ( k - 1 ) ( I i ( ℝ 2 ) ) ~ D ( k - 1 ) ( ℝ 2 )
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, D k ( I i ( ℝ 2 ) ) ~ D k ( ℝ 2 )
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, Cross ( I i Double ) ( T ( I i ( ℝ 2 ) ) )
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, Coercible ( I i Double ) ( I i ( ℝ 1 ) )
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)
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=> D k ( I i ( ℝ 2 ) ) ( I i ( ℝ 2 ) )
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-> ( D ( k - 1 ) ( I i ( ℝ 2 ) ) ( I i ( ℝ 1 ) )
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, D ( k - 2 ) ( I i ( ℝ 2 ) ) ( T ( I i ( ℝ 2 ) ) ) )
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instance CurveOrder 2 where
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line ( Segment a b :: Segment b ) =
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D \ ( coerce -> t ) ->
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D21 ( lerp @( T b ) t a b )
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( a --> b )
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origin
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bezier2 ( bez :: Quadratic.Bezier b ) =
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D \ ( coerce -> t ) ->
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D21 ( 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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bezier3 ( bez :: Cubic.Bezier b ) =
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D \ ( coerce -> t ) ->
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D21 ( 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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uncurryD = uncurryD2
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brushStroke ( D21 p p_t p_tt ) ( D22 b b_t b_s b_tt b_ts b_ss ) =
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D22 ( unT $ T p ^+^ T b )
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-- c = p + b
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( p_t ^+^ b_t ) b_s
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-- ∂c/∂t = dp/dt + ∂b/∂t
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-- ∂c/∂s = ∂b/∂s
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( p_tt ^+^ b_tt ) b_ts b_ss
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-- ∂²c/∂t² = d²p/dt² + ∂²b/∂t²
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-- ∂²c/∂t∂s = ∂²b/∂t∂s
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-- ∂²c/∂s² = ∂²b/∂s²
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envelopeEquation ( D22 _ c_t c_s c_tt c_ts c_ss ) =
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let !ee = c_t × c_s
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!ee_t = c_tt × c_s + c_t × c_ts
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!ee_s = c_ts × c_s + c_t × c_ss
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!𝛿E𝛿sdcdt = ee_s *^ c_t ^-^ ee_t *^ c_s
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in ( D12 ( coerce ee ) ( T $ coerce ee_t ) ( T $ coerce ee_s )
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, D0 𝛿E𝛿sdcdt )
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-- Computation of total derivative dc/dt:
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--
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-- dc/dt = ∂c/∂t + ∂c/∂s ∂s/∂t
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-- ∂s/∂t = - ∂E/∂t / ∂E/∂s
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--
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-- ∂E/∂s dc/dt = ∂E/∂s ∂c/∂t - ∂E/∂t ∂c/∂s.
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instance CurveOrder 3 where
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line ( Segment a b :: Segment b ) =
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D \ ( coerce -> t ) ->
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D31 ( lerp @( T b ) t a b )
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( a --> b )
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origin
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origin
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bezier2 ( bez :: Quadratic.Bezier b ) =
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D \ ( coerce -> t ) ->
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D31 ( 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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origin
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bezier3 ( bez :: Cubic.Bezier b ) =
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D \ ( coerce -> t ) ->
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D31 ( 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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( Cubic.bezier''' bez )
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uncurryD = uncurryD3
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brushStroke
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( D31 p p_t p_tt p_ttt )
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( D32 b b_t b_s b_tt b_ts b_ss b_ttt b_tts b_tss b_sss ) =
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D32
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( unT $ T p ^+^ T b )
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( p_t ^+^ b_t ) b_s
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( p_tt ^+^ b_tt ) b_ts b_ss
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( p_ttt ^+^ b_ttt ) b_tts b_tss b_sss
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envelopeEquation
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( D32 _ c_t c_s
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c_tt c_ts c_ss
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c_ttt c_tts c_tss c_sss )
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= let !ee = c_t × c_s
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!ee_t = c_tt × c_s + c_t × c_ts
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!ee_s = c_ts × c_s + c_t × c_ss
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!ee_tt = c_ttt × c_s
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+ c_tt × c_ts * 2
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+ c_t × c_tts
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!ee_ts = c_tts × c_s
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+ c_tt × c_ss
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-- + c_ts × c_ts -- cancels out
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+ c_t × c_tss
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!ee_ss = c_tss × c_s
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+ c_ts × c_ss * 2
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+ c_t × c_sss
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!𝛿E𝛿sdcdt = ee_s *^ c_t ^-^ ee_t *^ c_s
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!𝛿E𝛿sdcdt_t = ee_ts *^ c_t ^+^ ee_s *^ c_tt
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^-^ ( ee_tt *^ c_s ^+^ ee_t *^ c_ts )
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!𝛿E𝛿sdcdt_s = ee_ss *^ c_t ^+^ ee_s *^ c_ts
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^-^ ( ee_ts *^ c_s ^+^ ee_t *^ c_ss )
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in ( D22
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( coerce ee )
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( T $ coerce ee_t ) ( T $ coerce ee_s )
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( T $ coerce ee_tt) ( T $ coerce ee_ts ) ( T $ coerce ee_ss )
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, D12 𝛿E𝛿sdcdt ( T 𝛿E𝛿sdcdt_t ) ( T 𝛿E𝛿sdcdt_s ) )
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