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Merge pull request #148 from diagrams/projections-rebase
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Projections - non-affine transformations
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byorgey committed Feb 11, 2014
2 parents 36d87dd + e1404ab commit dad2704
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5 changes: 4 additions & 1 deletion diagrams-lib.cabal
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Expand Up @@ -44,6 +44,7 @@ Library
Diagrams.Solve,
Diagrams.Tangent,
Diagrams.Transform,
Diagrams.Deform
Diagrams.BoundingBox,
Diagrams.Names,
Diagrams.Envelope,
Expand All @@ -55,6 +56,7 @@ Library
Diagrams.TwoD.Arrow,
Diagrams.TwoD.Arrowheads,
Diagrams.TwoD.Combinators,
Diagrams.TwoD.Deform,
Diagrams.TwoD.Transform,
Diagrams.TwoD.Transform.ScaleInv,
Diagrams.TwoD.Ellipse,
Expand All @@ -73,12 +75,13 @@ Library
Diagrams.TwoD.Adjust,
Diagrams.ThreeD.Align,
Diagrams.ThreeD.Camera,
Diagrams.ThreeD.Deform,
Diagrams.ThreeD.Light,
Diagrams.ThreeD.Shapes,
Diagrams.ThreeD.Transform,
Diagrams.ThreeD.Types,
Diagrams.ThreeD.Vector,
Diagrams.ThreeD
Diagrams.ThreeD,
Diagrams.Animation,
Diagrams.Animation.Active,
Diagrams.Util,
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6 changes: 3 additions & 3 deletions src/Diagrams/Combinators.hs
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Expand Up @@ -160,10 +160,10 @@ deformEnvelope
:: ( Ord (Scalar v), Num (Scalar v), AdditiveGroup (Scalar v)
, Floating (Scalar v), HasLinearMap v, InnerSpace v, Monoid' m )
=> (Scalar v) -> v -> QDiagram b v m -> QDiagram b v m
deformEnvelope s v d = setEnvelope (getEnvelope d & _Wrapping Envelope %~ deform) d
deformEnvelope s v d = setEnvelope (getEnvelope d & _Wrapping Envelope %~ deformE) d
where
deform = Option . fmap deform' . getOption
deform' env v'
deformE = Option . fmap deformE' . getOption
deformE' env v'
| dot > 0 = Max $ getMax (env v') + (dot * s) / magnitude v'
| otherwise = env v'
where
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117 changes: 117 additions & 0 deletions src/Diagrams/Deform.hs
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@@ -0,0 +1,117 @@
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE UndecidableInstances #-}

module Diagrams.Deform (Deformation(..), Deformable(..), asDeformation) where

import Control.Lens (under, _Unwrapped)
import Data.AffineSpace
import Data.Basis
import Data.MemoTrie
import Data.Monoid hiding ((<>))
import Data.Semigroup
import Data.VectorSpace

import Diagrams.Core
import Diagrams.Located
import Diagrams.Parametric
import Diagrams.Path
import Diagrams.Segment
import Diagrams.Trail

------------------------------------------------------------
-- Deformations

-- | @Deformations@ are a superset of the affine transformations
-- represented by the 'Transformation' type. In general they are not
-- invertable. @Deformation@s include projective transformations.
-- @Deformation@ can represent other functions from points to points
-- which are "well-behaved", in that they do not introduce small wiggles.
data Deformation v = Deformation (Point v -> Point v)

instance Semigroup (Deformation v) where
(Deformation p1) <> (Deformation p2) = Deformation (p1 . p2)

instance Monoid (Deformation v) where
mappend = (<>)
mempty = Deformation id

class Deformable a where
-- | @deform' epsilon d a@ transforms @a@ by the deformation @d@.
-- If the type of @a@ is not closed under projection, approximate
-- to accuracy @epsilon@.
deform' :: Scalar (V a) -> Deformation (V a) -> a -> a

-- | @deform d a@ transforms @a@ by the deformation @d@.
-- If the type of @a@ is not closed under projection, @deform@
-- should call @deform'@ with some reasonable default value of
-- @epsilon@.
deform :: Deformation (V a) -> a -> a

-- | @asDeformation@ converts a 'Transformation' to a 'Deformation' by
-- discarding the inverse transform. This allows reusing
-- @Transformation@s in the construction of @Deformation@s.
asDeformation
:: ( HasTrie (Basis v), HasBasis v) => Transformation v -> Deformation v
asDeformation t = Deformation f' where
f' = papply t

------------------------------------------------------------
-- Instances

instance Deformable (Point v) where
deform' = const deform
deform (Deformation l) = l

-- | Cubic curves are not closed under perspective projections.
-- Therefore @Segment@s are not an instance of Deformable. However,
-- the deformation of a @Segment@ can be approximated to arbitrary
-- precision by a series of @Segment@s. @deformSegment@ does this,
-- which allows types built from lists of @Segment@s to themselves be
-- @Deformable@.
deformSegment :: (VectorSpace v, InnerSpace v, s ~ Scalar v, Ord s, Fractional s, Floating s) =>
s -> Deformation v -> FixedSegment v -> [FixedSegment v]
deformSegment epsilon t s
| goodEnough epsilon t s = [approx t s]
| otherwise = concatMap (deformSegment epsilon t) [s1, s2]
where
(s1, s2) = splitAtParam s 0.5

approx :: (VectorSpace v, InnerSpace v, s ~ Scalar v, Ord s, Fractional s, Floating s) =>
Deformation v -> FixedSegment v -> FixedSegment v
approx t (FLinear p0 p1) = FLinear (deform t p0) (deform t p1)
approx t (FCubic p0 c1 c2 p1) = FCubic (f p0) (f c1) (f c2) (f p1) where
f = deform t

goodEnough :: (VectorSpace v, InnerSpace v, s ~ Scalar v, Ord s, Fractional s, Floating s) =>
s -> Deformation v -> FixedSegment v -> Bool
goodEnough e t s =
all (< e) [magnitude $ deform t (s `atParam` u) .-. approx t s `atParam` u
| u <- [0.25, 0.5, 0.75]]

instance (VectorSpace v, InnerSpace v,
s ~ Scalar v, Ord s, Fractional s, Floating s, Show s, Show v) =>
Deformable (Located (Trail v)) where
deform' eps p t
| isLine $ unLoc t = line `at` p0
| otherwise = glueTrail line `at` p0
where
segs = concatMap (deformSegment eps p) $ fixTrail t
p0 = case segs of
(FLinear start _:_) -> start
(FCubic start _ _ _:_) -> start
_ -> loc t -- default in case of empty trail
line = trailFromSegments $ map (unLoc . fromFixedSeg) segs
deform p t = deform' (0.01 * extent) p t where
-- estimate the "size" of the Trail' as
-- the maximum distance to any vertex
extent = maximum . map dist . trailVertices $ t
dist pt = magnitude $ pt .-. loc t

instance (VectorSpace v, InnerSpace v,
s ~ Scalar v, Ord s, Fractional s, Floating s, Show s, Show v) =>
Deformable (Path v) where
deform' eps p = under _Unwrapped $ map (deform' eps p)
deform p = under _Unwrapped $ map (deform p)
5 changes: 5 additions & 0 deletions src/Diagrams/Prelude.hs
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Expand Up @@ -61,6 +61,10 @@ module Diagrams.Prelude
-- conjugation of transformations.
, module Diagrams.Transform

-- | Projective transformations and other deformations
-- lacking an inverse.
, module Diagrams.Deform

-- | Giving names to subdiagrams and later retrieving
-- subdiagrams by name.
, module Diagrams.Names
Expand Down Expand Up @@ -124,6 +128,7 @@ import Diagrams.Attributes
import Diagrams.Combinators
import Diagrams.Coordinates
import Diagrams.CubicSpline
import Diagrams.Deform
import Diagrams.Envelope
import Diagrams.Located
import Diagrams.Names
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29 changes: 29 additions & 0 deletions src/Diagrams/Segment.hs
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Expand Up @@ -375,6 +375,35 @@ instance VectorSpace v => Parametric (FixedSegment v) where

p3 = alerp p21 p22 t

instance Num (Scalar v) => DomainBounds (FixedSegment v)

instance (VectorSpace v, Num (Scalar v)) => EndValues (FixedSegment v) where
atStart (FLinear p0 _) = p0
atStart (FCubic p0 _ _ _) = p0
atEnd (FLinear _ p1) = p1
atEnd (FCubic _ _ _ p1 ) = p1

instance (VectorSpace v, Fractional (Scalar v)) => Sectionable (FixedSegment v) where
splitAtParam (FLinear p0 p1) t = (left, right)
where left = FLinear p0 p
right = FLinear p p1
p = alerp p0 p1 t
splitAtParam (FCubic p0 c1 c2 p1) t = (left, right)
where left = FCubic p0 a b cut
right = FCubic cut c d p1
-- first round
a = alerp p0 c1 t
p = alerp c1 c2 t
d = alerp c2 p1 t
-- second round
b = alerp a p t
c = alerp p d t
-- final round
cut = alerp b c t

reverseDomain (FLinear p0 p1) = FLinear p1 p0
reverseDomain (FCubic p0 c1 c2 p1) = FCubic p1 c2 c1 p0

------------------------------------------------------------
-- Segment measures --------------------------------------
------------------------------------------------------------
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51 changes: 51 additions & 0 deletions src/Diagrams/ThreeD/Deform.hs
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@@ -0,0 +1,51 @@
module Diagrams.ThreeD.Deform where

import Control.Lens

import Diagrams.Deform

import Diagrams.Coordinates
import Diagrams.ThreeD.Types

-- | The parallel projection onto the plane x=0
parallelX0 :: Deformation R3
parallelX0 = Deformation (& _x .~ 0)

-- | The perspective division onto the plane x=1 along lines going
-- through the origin.
perspectiveX1 :: Deformation R3
perspectiveX1 = Deformation (\p -> let x = p^._x in
p & _x .~ 1 & _y //~ x & _z //~ x )

-- | The parallel projection onto the plane y=0
parallelY0 :: Deformation R3
parallelY0 = Deformation (& _y .~ 0)

-- | The perspective division onto the plane y=1 along lines going
-- through the origin.
perspectiveY1 :: Deformation R3
perspectiveY1 = Deformation (\p -> let y = p^._y in
p & _x //~ y & _y .~ 1 & _z //~ y )

-- | The parallel projection onto the plane z=0
parallelZ0 :: Deformation R3
parallelZ0 = Deformation (& _z .~ 0)

-- | The perspective division onto the plane z=1 along lines going
-- through the origin.
perspectiveZ1 :: Deformation R3
perspectiveZ1 = Deformation (\p -> let z = p^._z in
p & _x //~ z & _y //~ z & _z .~ 1 )

-- | The viewing transform for a viewer facing along the positive X
-- axis. X coördinates stay fixed, while Y coördinates are compressed
-- with increasing distance. @asDeformation (translation unitX) <>
-- parallelX0 <> frustrumX = perspectiveX1@
facingX :: Deformation R3
facingX = Deformation (\v -> v & _y //~ (v^._x) & _z //~ (v^._x))

facingY :: Deformation R3
facingY = Deformation (\v -> v & _x //~ (v^._y) & _z //~ (v^._y))

facingZ :: Deformation R3
facingZ = Deformation (\v -> v & _x //~ (v^._z) & _y //~ (v^._z))
5 changes: 5 additions & 0 deletions src/Diagrams/TwoD.hs
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Expand Up @@ -189,6 +189,10 @@ module Diagrams.TwoD
, shearingX, shearX
, shearingY, shearY

-- * Deformations - non-affine transforms
, parallelX0, perspectiveX1, parallelY0, perspectiveY1
, facingX, facingY

-- * Combinators
-- ** Combining multiple diagrams
, (===), (|||), atAngle
Expand Down Expand Up @@ -247,6 +251,7 @@ import Diagrams.TwoD.Image
import Diagrams.TwoD.Model
import Diagrams.TwoD.Path
import Diagrams.TwoD.Polygons
import Diagrams.TwoD.Deform
import Diagrams.TwoD.Shapes
import Diagrams.TwoD.Size
import Diagrams.TwoD.Text
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36 changes: 36 additions & 0 deletions src/Diagrams/TwoD/Deform.hs
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@@ -0,0 +1,36 @@
module Diagrams.TwoD.Deform where

import Control.Lens

import Diagrams.Deform

import Diagrams.Coordinates
import Diagrams.TwoD.Types

-- | The parallel projection onto the line x=0
parallelX0 :: Deformation R2
parallelX0 = Deformation (& _x .~ 0)

-- | The perspective division onto the line x=1 along lines going
-- through the origin.
perspectiveX1 :: Deformation R2
perspectiveX1 = Deformation (\p -> p & _y //~ (p^._x) & _x .~ 1)

-- | The parallel projection onto the line y=0
parallelY0 :: Deformation R2
parallelY0 = Deformation (& _y .~ 0)

-- | The perspective division onto the line y=1 along lines going
-- through the origin.
perspectiveY1 :: Deformation R2
perspectiveY1 = Deformation (\p -> p & _x //~ (p^._y) & _y .~ 1)

-- | The viewing transform for a viewer facing along the positive X
-- axis. X coördinates stay fixed, while Y coördinates are compressed
-- with increasing distance. @asDeformation (translation unitX) <>
-- parallelX0 <> frustrumX = perspectiveX1@
facingX :: Deformation R2
facingX = Deformation (\v -> v & _y //~ (v^._x))

facingY :: Deformation R2
facingY = Deformation (\v -> v & _x //~ (v^._y))
4 changes: 4 additions & 0 deletions test/stretchtest.svg
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