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compute roots of envelope equation
This commit is contained in:
sheaf
parent
df2469487d
commit
7431e8ba67
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@ -83,6 +83,7 @@ library
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exposed-modules:
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exposed-modules:
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Math.Bezier.Cubic
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Math.Bezier.Cubic
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, Math.Bezier.Cubic.Fit
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, Math.Bezier.Cubic.Fit
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, Math.Bezier.Envelope
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, Math.Bezier.Quadratic
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, Math.Bezier.Quadratic
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, Math.Bezier.Spline
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, Math.Bezier.Spline
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, Math.Bezier.Stroke
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, Math.Bezier.Stroke
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455
src/lib/Math/Bezier/Envelope.hs
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455
src/lib/Math/Bezier/Envelope.hs
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@ -0,0 +1,455 @@
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{-# LANGUAGE ScopedTypeVariables #-}
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{-# LANGUAGE TypeApplications #-}
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module Math.Bezier.Envelope where
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-- acts
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import Data.Act
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( Torsor((-->)) )
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-- deepseq
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import Control.DeepSeq
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( NFData )
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-- primitive
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import Data.Primitive.Types
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( Prim )
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-- MetaBrush
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import Math.Roots
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( realRoots )
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import qualified Math.Bezier.Cubic as Cubic
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( Bezier(..), bezier, bezier' )
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import qualified Math.Bezier.Quadratic as Quadratic
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( Bezier(..), bezier, bezier' )
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import Math.Module
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( Module((^+^),(*^)), lerp, cross )
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import Math.Vector2D
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( Point2D(..), Vector2D(..), Segment(..) )
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--------------------------------------------------------------------------------
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-- | Find the roots of the envelope equation for a family of cubic Bézier curves
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-- varying along a cubic Bézier path.
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--
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-- \[ c(t,u) = p(t) + b(t,u), \]
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--
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-- where \( t \mapsto p(t) \) describes the underlying path,
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-- and \( u \mapsto b(t_0,u) \) describes the brush shape at point \( t = t_0 \).
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--
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-- The envelope equation is then:
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--
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-- \[ \frac{\partial c}{\partial t} \cross \frac{\partial c}{\partial u} = 0. \]
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--
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-- Given \( t_0 \), this function returns a (possibly empty) list of values \( u_i \)
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-- satisfying the envelope equation at \( (t_0, u_i) \).
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--
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-- The points \( c(t_0,u_i) \) are thus potential outline points on the contour stroked
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-- by the brush as it moves along the path.
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envelope33
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:: forall r
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. ( RealFloat r, Prim r, NFData r )
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=> Cubic.Bezier ( Point2D r ) -> Cubic.Bezier ( Cubic.Bezier ( Point2D r ) ) -> r -> [ r ]
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envelope33 path
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( Cubic.Bezier
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( Cubic.Bezier b00 b01 b02 b03 )
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( Cubic.Bezier b10 b11 b12 b13 )
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( Cubic.Bezier b20 b21 b22 b23 )
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( Cubic.Bezier b30 b31 b32 b33 )
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) t0 = realRoots 50 [ a5, a4, a3, a2, a1, a0 ]
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where
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-- Compute ∂p/∂t(t0).
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dpdt :: Vector2D r
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dpdt = Cubic.bezier' @( Vector2D r ) path t0
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-- Compute ∂b/∂t(t0,u) using the Bernstein basis:
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--
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-- ∂b/∂t(t0,u) = Cubic.bezier ( Cubic.Bezier ct0 ct1 ct2 ct3 ) u.
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ct0, ct1, ct2, ct3, dt0, dt1, dt2, dt3 :: Vector2D r
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ct0 = Cubic.bezier' @( Vector2D r ) ( Cubic.Bezier b00 b10 b20 b30 ) t0
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ct1 = Cubic.bezier' @( Vector2D r ) ( Cubic.Bezier b01 b11 b21 b31 ) t0
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ct2 = Cubic.bezier' @( Vector2D r ) ( Cubic.Bezier b02 b12 b22 b32 ) t0
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ct3 = Cubic.bezier' @( Vector2D r ) ( Cubic.Bezier b03 b13 b23 b33 ) t0
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-- Add ∂p/∂t and convert the Bernstein representation to the monomial basis to obtain
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--
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-- ∂c/∂t(t0,u) = dt0 + u dt1 + u² dt2 + u³ dt3.
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dt0 = ct0 ^+^ dpdt
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dt1 = 3 *^ ( ct0 --> ct1 )
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dt2 = 3 *^ ( ct1 --> ct0 ^+^ ct1 --> ct2 )
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dt3 = ct0 --> ct3 ^+^ 3 *^ ( ct2 --> ct1 )
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-- Compute ∂c/∂u(t0,u) using the Bernstein basis:
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--
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-- ∂c/∂u(t0,u) = Cubic.bezier' ( Cubic.Bezier cu0 cu1 cu2 cu3 ) u.
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cu0, cu1, cu2, cu3 :: Point2D r
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cu0 = Cubic.bezier @( Vector2D r ) ( Cubic.Bezier b00 b10 b20 b30 ) t0
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cu1 = Cubic.bezier @( Vector2D r ) ( Cubic.Bezier b01 b11 b21 b31 ) t0
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cu2 = Cubic.bezier @( Vector2D r ) ( Cubic.Bezier b02 b12 b22 b32 ) t0
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cu3 = Cubic.bezier @( Vector2D r ) ( Cubic.Bezier b03 b13 b23 b33 ) t0
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-- Convert the Bernstein representation to the monomial basis to obtain
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--
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-- ∂c/∂u(t0,u) = du0 + u du1 + u² du2.
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du0, du1, du2 :: Vector2D r
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du0 = cu0 --> cu1
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du1 = 2 *^ ( cu1 --> cu0 ^+^ cu1 --> cu2 )
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du2 = cu0 --> cu3 ^+^ 3 *^ ( cu2 --> cu1 )
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-- Expand out the cross-product ∂c/∂t × ∂c/∂u to obtain the envelope equation:
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--
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-- a0 + a1 u + a2 u² + a3 u³ + a4 u⁴ + a5 u⁵.
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a0, a1, a2, a3, a4, a5 :: r
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a0 = dt0 `cross` du0
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a1 = dt1 `cross` du0 + dt0 `cross` du1
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a2 = dt2 `cross` du0 + dt1 `cross` du1 + dt0 `cross` du2
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a3 = dt3 `cross` du0 + dt2 `cross` du1 + dt1 `cross` du2
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a4 = dt3 `cross` du1 + dt2 `cross` du2
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a5 = dt3 `cross` du2
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-- | Find the roots of the envelope equation for a family of cubic Bézier curves
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-- varying along a quadratic Bézier path.
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--
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-- See 'envelope33' for more information.
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envelope23
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:: forall r
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. ( RealFloat r, Prim r, NFData r )
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=> Quadratic.Bezier ( Point2D r ) -> Quadratic.Bezier ( Cubic.Bezier ( Point2D r ) ) -> r -> [ r ]
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envelope23 path
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( Quadratic.Bezier
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( Cubic.Bezier b00 b01 b02 b03 )
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( Cubic.Bezier b10 b11 b12 b13 )
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( Cubic.Bezier b20 b21 b22 b23 )
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) t0 = realRoots 50 [ a5, a4, a3, a2, a1, a0 ]
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where
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dpdt :: Vector2D r
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dpdt = Quadratic.bezier' @( Vector2D r ) path t0
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ct0, ct1, ct2, ct3, dt0, dt1, dt2, dt3 :: Vector2D r
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ct0 = Quadratic.bezier' @( Vector2D r ) ( Quadratic.Bezier b00 b10 b20 ) t0
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ct1 = Quadratic.bezier' @( Vector2D r ) ( Quadratic.Bezier b01 b11 b21 ) t0
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ct2 = Quadratic.bezier' @( Vector2D r ) ( Quadratic.Bezier b02 b12 b22 ) t0
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ct3 = Quadratic.bezier' @( Vector2D r ) ( Quadratic.Bezier b03 b13 b23 ) t0
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dt0 = ct0 ^+^ dpdt
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dt1 = 3 *^ ( ct0 --> ct1 )
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dt2 = 3 *^ ( ct1 --> ct0 ^+^ ct1 --> ct2 )
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dt3 = ct0 --> ct3 ^+^ 3 *^ ( ct2 --> ct1 )
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cu0, cu1, cu2, cu3 :: Point2D r
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cu0 = Quadratic.bezier @( Vector2D r ) ( Quadratic.Bezier b00 b10 b20 ) t0
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cu1 = Quadratic.bezier @( Vector2D r ) ( Quadratic.Bezier b01 b11 b21 ) t0
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cu2 = Quadratic.bezier @( Vector2D r ) ( Quadratic.Bezier b02 b12 b22 ) t0
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cu3 = Quadratic.bezier @( Vector2D r ) ( Quadratic.Bezier b03 b13 b23 ) t0
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du0, du1, du2 :: Vector2D r
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du0 = cu0 --> cu1
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du1 = 2 *^ ( cu1 --> cu0 ^+^ cu1 --> cu2 )
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du2 = cu0 --> cu3 ^+^ 3 *^ ( cu2 --> cu1 )
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a0, a1, a2, a3, a4, a5 :: r
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a0 = dt0 `cross` du0
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a1 = dt1 `cross` du0 + dt0 `cross` du1
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a2 = dt2 `cross` du0 + dt1 `cross` du1 + dt0 `cross` du2
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a3 = dt3 `cross` du0 + dt2 `cross` du1 + dt1 `cross` du2
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a4 = dt3 `cross` du1 + dt2 `cross` du2
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a5 = dt3 `cross` du2
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-- | Find the roots of the envelope equation for a family of cubic Bézier curves
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-- varying along a straight line path.
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--
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-- See 'envelope33' for more information.
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envelope13
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:: forall r
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. ( RealFloat r, Prim r, NFData r )
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=> Segment ( Point2D r ) -> Segment ( Cubic.Bezier ( Point2D r ) ) -> r -> [ r ]
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envelope13 ( Segment p0 p1 )
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( Segment
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( Cubic.Bezier b00 b01 b02 b03 )
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( Cubic.Bezier b10 b11 b12 b13 )
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) t0 = realRoots 50 [ a5, a4, a3, a2, a1, a0 ]
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where
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dpdt :: Vector2D r
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dpdt = p0 --> p1
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ct0, ct1, ct2, ct3, dt0, dt1, dt2, dt3 :: Vector2D r
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ct0 = b00 --> b10
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ct1 = b01 --> b11
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ct2 = b02 --> b12
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ct3 = b03 --> b13
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dt0 = ct0 ^+^ dpdt
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dt1 = 3 *^ ( ct0 --> ct1 )
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dt2 = 3 *^ ( ct1 --> ct0 ^+^ ct1 --> ct2 )
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dt3 = ct0 --> ct3 ^+^ 3 *^ ( ct2 --> ct1 )
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cu0, cu1, cu2, cu3 :: Point2D r
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cu0 = lerp @( Vector2D r ) t0 b00 b10
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cu1 = lerp @( Vector2D r ) t0 b01 b11
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cu2 = lerp @( Vector2D r ) t0 b02 b12
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cu3 = lerp @( Vector2D r ) t0 b03 b13
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du0, du1, du2 :: Vector2D r
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du0 = cu0 --> cu1
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du1 = 2 *^ ( cu1 --> cu0 ^+^ cu1 --> cu2 )
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du2 = cu0 --> cu3 ^+^ 3 *^ ( cu2 --> cu1 )
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a0, a1, a2, a3, a4, a5 :: r
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a0 = dt0 `cross` du0
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a1 = dt1 `cross` du0 + dt0 `cross` du1
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a2 = dt2 `cross` du0 + dt1 `cross` du1 + dt0 `cross` du2
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a3 = dt3 `cross` du0 + dt2 `cross` du1 + dt1 `cross` du2
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a4 = dt3 `cross` du1 + dt2 `cross` du2
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a5 = dt3 `cross` du2
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-- | Find the roots of the envelope equation for a family of quadratic Bézier curves
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-- varying along a cubic Bézier path.
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--
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-- See 'envelope33' for more information.
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envelope32
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:: forall r
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. ( RealFloat r, Prim r, NFData r )
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=> Cubic.Bezier ( Point2D r ) -> Cubic.Bezier ( Quadratic.Bezier ( Point2D r ) ) -> r -> [ r ]
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envelope32 path
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( Cubic.Bezier
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( Quadratic.Bezier b00 b01 b02 )
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( Quadratic.Bezier b10 b11 b12 )
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( Quadratic.Bezier b20 b21 b22 )
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( Quadratic.Bezier b30 b31 b32 )
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) t0 = realRoots 50 [ a3, a2, a1, a0 ]
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where
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dpdt :: Vector2D r
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dpdt = Cubic.bezier' @( Vector2D r ) path t0
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ct0, ct1, ct2, dt0, dt1, dt2 :: Vector2D r
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ct0 = Cubic.bezier' @( Vector2D r ) ( Cubic.Bezier b00 b10 b20 b30 ) t0
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ct1 = Cubic.bezier' @( Vector2D r ) ( Cubic.Bezier b01 b11 b21 b31 ) t0
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ct2 = Cubic.bezier' @( Vector2D r ) ( Cubic.Bezier b02 b12 b22 b32 ) t0
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dt0 = ct0 ^+^ dpdt
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dt1 = 2 *^ ( ct0 --> ct1 )
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dt2 = ct1 --> ct0 ^+^ ct1 --> ct2
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cu0, cu1, cu2 :: Point2D r
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cu0 = Cubic.bezier @( Vector2D r ) ( Cubic.Bezier b00 b10 b20 b30 ) t0
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cu1 = Cubic.bezier @( Vector2D r ) ( Cubic.Bezier b01 b11 b21 b31 ) t0
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cu2 = Cubic.bezier @( Vector2D r ) ( Cubic.Bezier b02 b12 b22 b32 ) t0
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du0, du1 :: Vector2D r
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du0 = cu0 --> cu1
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du1 = cu1 --> cu0 ^+^ cu1 --> cu2
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a0, a1, a2, a3 :: r
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a0 = dt0 `cross` du0
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a1 = dt1 `cross` du0 + dt0 `cross` du1
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a2 = dt2 `cross` du0 + dt1 `cross` du1
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a3 = dt2 `cross` du1
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-- | Find the roots of the envelope equation for a family of quadratic Bézier curves
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-- varying along a quadratic Bézier path.
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--
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-- See 'envelope33' for more information.
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envelope22
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:: forall r
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. ( RealFloat r, Prim r, NFData r )
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=> Quadratic.Bezier ( Point2D r ) -> Quadratic.Bezier ( Quadratic.Bezier ( Point2D r ) ) -> r -> [ r ]
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envelope22 path
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( Quadratic.Bezier
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( Quadratic.Bezier b00 b01 b02 )
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( Quadratic.Bezier b10 b11 b12 )
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( Quadratic.Bezier b20 b21 b22 )
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) t0 = realRoots 50 [ a3, a2, a1, a0 ]
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where
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dpdt :: Vector2D r
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dpdt = Quadratic.bezier' @( Vector2D r ) path t0
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ct0, ct1, ct2, dt0, dt1, dt2 :: Vector2D r
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ct0 = Quadratic.bezier' @( Vector2D r ) ( Quadratic.Bezier b00 b10 b20 ) t0
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ct1 = Quadratic.bezier' @( Vector2D r ) ( Quadratic.Bezier b01 b11 b21 ) t0
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ct2 = Quadratic.bezier' @( Vector2D r ) ( Quadratic.Bezier b02 b12 b22 ) t0
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dt0 = ct0 ^+^ dpdt
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dt1 = 2 *^ ( ct0 --> ct1 )
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dt2 = ct1 --> ct0 ^+^ ct1 --> ct2
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cu0, cu1, cu2 :: Point2D r
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cu0 = Quadratic.bezier @( Vector2D r ) ( Quadratic.Bezier b00 b10 b20 ) t0
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cu1 = Quadratic.bezier @( Vector2D r ) ( Quadratic.Bezier b01 b11 b21 ) t0
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cu2 = Quadratic.bezier @( Vector2D r ) ( Quadratic.Bezier b02 b12 b22 ) t0
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du0, du1 :: Vector2D r
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du0 = cu0 --> cu1
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du1 = cu1 --> cu0 ^+^ cu1 --> cu2
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a0, a1, a2, a3 :: r
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a0 = dt0 `cross` du0
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a1 = dt1 `cross` du0 + dt0 `cross` du1
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a2 = dt2 `cross` du0 + dt1 `cross` du1
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a3 = dt2 `cross` du1
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-- | Find the roots of the envelope equation for a family of quadratic Bézier curves
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-- varying along a straight line.
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--
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-- See 'envelope33' for more information.
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||||||
|
envelope12
|
||||||
|
:: forall r
|
||||||
|
. ( RealFloat r, Prim r, NFData r )
|
||||||
|
=> Segment ( Point2D r ) -> Segment ( Quadratic.Bezier ( Point2D r ) ) -> r -> [ r ]
|
||||||
|
envelope12 ( Segment p0 p1 )
|
||||||
|
( Segment
|
||||||
|
( Quadratic.Bezier b00 b01 b02 )
|
||||||
|
( Quadratic.Bezier b10 b11 b12 )
|
||||||
|
) t0 = realRoots 50 [ a3, a2, a1, a0 ]
|
||||||
|
|
||||||
|
where
|
||||||
|
|
||||||
|
dpdt :: Vector2D r
|
||||||
|
dpdt = p0 --> p1
|
||||||
|
|
||||||
|
ct0, ct1, ct2, dt0, dt1, dt2 :: Vector2D r
|
||||||
|
ct0 = b00 --> b10
|
||||||
|
ct1 = b01 --> b11
|
||||||
|
ct2 = b02 --> b12
|
||||||
|
|
||||||
|
dt0 = ct0 ^+^ dpdt
|
||||||
|
dt1 = 2 *^ ( ct0 --> ct1 )
|
||||||
|
dt2 = ct1 --> ct0 ^+^ ct1 --> ct2
|
||||||
|
|
||||||
|
cu0, cu1, cu2 :: Point2D r
|
||||||
|
cu0 = lerp @( Vector2D r ) t0 b00 b10
|
||||||
|
cu1 = lerp @( Vector2D r ) t0 b01 b11
|
||||||
|
cu2 = lerp @( Vector2D r ) t0 b02 b12
|
||||||
|
|
||||||
|
du0, du1 :: Vector2D r
|
||||||
|
du0 = cu0 --> cu1
|
||||||
|
du1 = cu1 --> cu0 ^+^ cu1 --> cu2
|
||||||
|
|
||||||
|
a0, a1, a2, a3 :: r
|
||||||
|
a0 = dt0 `cross` du0
|
||||||
|
a1 = dt1 `cross` du0 + dt0 `cross` du1
|
||||||
|
a2 = dt2 `cross` du0 + dt1 `cross` du1
|
||||||
|
a3 = dt2 `cross` du1
|
||||||
|
|
||||||
|
-- | Find the roots of the envelope equation for a family of line segments
|
||||||
|
-- varying along a cubic Bézier curve.
|
||||||
|
--
|
||||||
|
-- See 'envelope33' for more information.
|
||||||
|
envelope31
|
||||||
|
:: forall r
|
||||||
|
. ( RealFloat r, Prim r, NFData r )
|
||||||
|
=> Cubic.Bezier ( Point2D r ) -> Cubic.Bezier ( Segment ( Point2D r ) ) -> r -> [ r ]
|
||||||
|
envelope31 path
|
||||||
|
( Cubic.Bezier
|
||||||
|
( Segment b00 b01 )
|
||||||
|
( Segment b10 b11 )
|
||||||
|
( Segment b20 b21 )
|
||||||
|
( Segment b30 b31 )
|
||||||
|
) t0 = [ - a1 / a0 ]
|
||||||
|
|
||||||
|
where
|
||||||
|
|
||||||
|
dpdt :: Vector2D r
|
||||||
|
dpdt = Cubic.bezier' @( Vector2D r ) path t0
|
||||||
|
|
||||||
|
ct0, ct1, dt0, dt1 :: Vector2D r
|
||||||
|
ct0 = Cubic.bezier' @( Vector2D r ) ( Cubic.Bezier b00 b10 b20 b30 ) t0
|
||||||
|
ct1 = Cubic.bezier' @( Vector2D r ) ( Cubic.Bezier b01 b11 b21 b31 ) t0
|
||||||
|
|
||||||
|
dt0 = ct0 ^+^ dpdt
|
||||||
|
dt1 = ct0 --> ct1
|
||||||
|
|
||||||
|
cu0, cu1 :: Point2D r
|
||||||
|
cu0 = Cubic.bezier @( Vector2D r ) ( Cubic.Bezier b00 b10 b20 b30 ) t0
|
||||||
|
cu1 = Cubic.bezier @( Vector2D r ) ( Cubic.Bezier b01 b11 b21 b31 ) t0
|
||||||
|
|
||||||
|
du0 :: Vector2D r
|
||||||
|
du0 = cu0 --> cu1
|
||||||
|
|
||||||
|
a0, a1 :: r
|
||||||
|
a0 = dt0 `cross` du0
|
||||||
|
a1 = dt1 `cross` du0
|
||||||
|
|
||||||
|
-- | Find the roots of the envelope equation for a family of line segments
|
||||||
|
-- varying along a quadratic Bézier curve.
|
||||||
|
--
|
||||||
|
-- See 'envelope33' for more information.
|
||||||
|
envelope21
|
||||||
|
:: forall r
|
||||||
|
. ( RealFloat r, Prim r, NFData r )
|
||||||
|
=> Quadratic.Bezier ( Point2D r ) -> Quadratic.Bezier ( Segment ( Point2D r ) ) -> r -> [ r ]
|
||||||
|
envelope21 path
|
||||||
|
( Quadratic.Bezier
|
||||||
|
( Segment b00 b01 )
|
||||||
|
( Segment b10 b11 )
|
||||||
|
( Segment b20 b21 )
|
||||||
|
) t0 = [ - a1 / a0 ]
|
||||||
|
|
||||||
|
where
|
||||||
|
|
||||||
|
dpdt :: Vector2D r
|
||||||
|
dpdt = Quadratic.bezier' @( Vector2D r ) path t0
|
||||||
|
|
||||||
|
ct0, ct1, dt0, dt1 :: Vector2D r
|
||||||
|
ct0 = Quadratic.bezier' @( Vector2D r ) ( Quadratic.Bezier b00 b10 b20 ) t0
|
||||||
|
ct1 = Quadratic.bezier' @( Vector2D r ) ( Quadratic.Bezier b01 b11 b21 ) t0
|
||||||
|
|
||||||
|
dt0 = ct0 ^+^ dpdt
|
||||||
|
dt1 = ct0 --> ct1
|
||||||
|
|
||||||
|
cu0, cu1 :: Point2D r
|
||||||
|
cu0 = Quadratic.bezier @( Vector2D r ) ( Quadratic.Bezier b00 b10 b20 ) t0
|
||||||
|
cu1 = Quadratic.bezier @( Vector2D r ) ( Quadratic.Bezier b01 b11 b21 ) t0
|
||||||
|
|
||||||
|
du0 :: Vector2D r
|
||||||
|
du0 = cu0 --> cu1
|
||||||
|
|
||||||
|
a0, a1 :: r
|
||||||
|
a0 = dt0 `cross` du0
|
||||||
|
a1 = dt1 `cross` du0
|
||||||
|
|
||||||
|
-- | Find the roots of the envelope equation for a family of line segments
|
||||||
|
-- varying along a straight line path.
|
||||||
|
--
|
||||||
|
-- See 'envelope33' for more information.
|
||||||
|
envelope11
|
||||||
|
:: forall r
|
||||||
|
. ( RealFloat r, Prim r, NFData r )
|
||||||
|
=> Segment ( Point2D r ) -> Segment ( Segment ( Point2D r ) ) -> r -> [ r ]
|
||||||
|
envelope11 ( Segment p0 p1 )
|
||||||
|
( Segment
|
||||||
|
( Segment b00 b01 )
|
||||||
|
( Segment b10 b11 )
|
||||||
|
) t0 = [ - a1 / a0 ]
|
||||||
|
|
||||||
|
where
|
||||||
|
|
||||||
|
dpdt :: Vector2D r
|
||||||
|
dpdt = p0 --> p1
|
||||||
|
|
||||||
|
ct0, ct1, dt0, dt1 :: Vector2D r
|
||||||
|
ct0 = b00 --> b10
|
||||||
|
ct1 = b01 --> b11
|
||||||
|
|
||||||
|
dt0 = ct0 ^+^ dpdt
|
||||||
|
dt1 = ct0 --> ct1
|
||||||
|
|
||||||
|
cu0, cu1 :: Point2D r
|
||||||
|
cu0 = lerp @( Vector2D r ) t0 b00 b10
|
||||||
|
cu1 = lerp @( Vector2D r ) t0 b01 b11
|
||||||
|
|
||||||
|
du0 :: Vector2D r
|
||||||
|
du0 = cu0 --> cu1
|
||||||
|
|
||||||
|
a0, a1 :: r
|
||||||
|
a0 = dt0 `cross` du0
|
||||||
|
a1 = dt1 `cross` du0
|
||||||
Loading…
Reference in a new issue