{-# LANGUAGE FlexibleContexts , FlexibleInstances , TypeFamilies , ViewPatterns #-} ----------------------------------------------------------------------------- -- | -- Module : Diagrams.TwoD.Transform -- Copyright : (c) 2011 diagrams-lib team (see LICENSE) -- License : BSD-style (see LICENSE) -- Maintainer : diagrams-discuss@googlegroups.com -- -- Transformations specific to two dimensions, with a few generic -- transformations (uniform scaling, translation) also re-exported for -- convenience. -- ----------------------------------------------------------------------------- module Diagrams.TwoD.Transform ( -- * Rotation rotation, rotate, rotateBy , rotationAbout, rotateAbout -- * Scaling , scalingX, scaleX , scalingY, scaleY , scaling, scale , scaleToX, scaleToY , scaleUToX, scaleUToY -- * Translation , translationX, translateX , translationY, translateY , translation, translate -- * Reflection , reflectionX, reflectX , reflectionY, reflectY , reflectionAbout, reflectAbout -- * Shears , shearingX, shearX , shearingY, shearY -- * Scale invariance , ScaleInv(..), scaleInv ) where import Diagrams.Core import Control.Newtype (over) import Diagrams.Coordinates import Diagrams.Transform import Diagrams.TwoD.Size (width, height) import Diagrams.TwoD.Types import Diagrams.TwoD.Vector (direction) import Data.Semigroup import Data.AffineSpace import Control.Arrow (first, second) -- Rotation ------------------------------------------------ -- | Create a transformation which performs a rotation about the local -- origin by the given angle. See also 'rotate'. rotation :: Angle a => a -> T2 rotation ang = fromLinear r (linv r) where r = rot theta <-> rot (-theta) Rad theta = convertAngle ang rot th (coords -> x :& y) = (cos th * x - sin th * y) & (sin th * x + cos th * y) -- | Rotate about the local origin by the given angle. Positive angles -- correspond to counterclockwise rotation, negative to -- clockwise. The angle can be expressed using any type which is an -- instance of 'Angle'. For example, @rotate (1\/4 :: -- 'CircleFrac')@, @rotate (tau\/4 :: 'Rad')@, and @rotate (90 :: -- 'Deg')@ all represent the same transformation, namely, a -- counterclockwise rotation by a right angle. To rotate about some -- point other than the local origin, see 'rotateAbout'. -- -- Note that writing @rotate (1\/4)@, with no type annotation, will -- yield an error since GHC cannot figure out which sort of angle -- you want to use. In this common situation you can use -- 'rotateBy', which is specialized to take a 'CircleFrac' argument. rotate :: (Transformable t, V t ~ R2, Angle a) => a -> t -> t rotate = transform . rotation -- | A synonym for 'rotate', specialized to only work with -- @CircleFrac@ arguments; it can be more convenient to write -- @rotateBy (1\/4)@ than @'rotate' (1\/4 :: 'CircleFrac')@. rotateBy :: (Transformable t, V t ~ R2) => CircleFrac -> t -> t rotateBy = transform . rotation -- | @rotationAbout p@ is a rotation about the point @p@ (instead of -- around the local origin). rotationAbout :: Angle a => P2 -> a -> T2 rotationAbout p angle = conjugate (translation (origin .-. p)) (rotation angle) -- | @rotateAbout p@ is like 'rotate', except it rotates around the -- point @p@ instead of around the local origin. rotateAbout :: (Transformable t, V t ~ R2, Angle a) => P2 -> a -> t -> t rotateAbout p angle = rotate angle `under` translation (origin .-. p) -- Scaling ------------------------------------------------- -- | Construct a transformation which scales by the given factor in -- the x (horizontal) direction. scalingX :: Double -> T2 scalingX c = fromLinear s s where s = (over r2 . first) (*c) <-> (over r2 . first) (/c) -- | Scale a diagram by the given factor in the x (horizontal) -- direction. To scale uniformly, use 'scale'. scaleX :: (Transformable t, V t ~ R2) => Double -> t -> t scaleX = transform . scalingX -- | Construct a transformation which scales by the given factor in -- the y (vertical) direction. scalingY :: Double -> T2 scalingY c = fromLinear s s where s = (over r2 . second) (*c) <-> (over r2 . second) (/c) -- | Scale a diagram by the given factor in the y (vertical) -- direction. To scale uniformly, use 'scale'. scaleY :: (Transformable t, V t ~ R2) => Double -> t -> t scaleY = transform . scalingY -- | @scaleToX w@ scales a diagram in the x (horizontal) direction by -- whatever factor required to make its width @w@. @scaleToX@ -- should not be applied to diagrams with a width of 0, such as -- 'vrule'. scaleToX :: (Enveloped t, Transformable t, V t ~ R2) => Double -> t -> t scaleToX w d = scaleX (w / width d) d -- | @scaleToY h@ scales a diagram in the y (vertical) direction by -- whatever factor required to make its height @h@. @scaleToY@ -- should not be applied to diagrams with a height of 0, such as -- 'hrule'. scaleToY :: (Enveloped t, Transformable t, V t ~ R2) => Double -> t -> t scaleToY h d = scaleY (h / height d) d -- | @scaleUToX w@ scales a diagram /uniformly/ by whatever factor -- required to make its width @w@. @scaleUToX@ should not be -- applied to diagrams with a width of 0, such as 'vrule'. scaleUToX :: (Enveloped t, Transformable t, V t ~ R2) => Double -> t -> t scaleUToX w d = scale (w / width d) d -- | @scaleUToY h@ scales a diagram /uniformly/ by whatever factor -- required to make its height @h@. @scaleUToY@ should not be applied -- to diagrams with a height of 0, such as 'hrule'. scaleUToY :: (Enveloped t, Transformable t, V t ~ R2) => Double -> t -> t scaleUToY h d = scale (h / height d) d -- Translation --------------------------------------------- -- | Construct a transformation which translates by the given distance -- in the x (horizontal) direction. translationX :: Double -> T2 translationX x = translation (x & 0) -- | Translate a diagram by the given distance in the x (horizontal) -- direction. translateX :: (Transformable t, V t ~ R2) => Double -> t -> t translateX = transform . translationX -- | Construct a transformation which translates by the given distance -- in the y (vertical) direction. translationY :: Double -> T2 translationY y = translation (0 & y) -- | Translate a diagram by the given distance in the y (vertical) -- direction. translateY :: (Transformable t, V t ~ R2) => Double -> t -> t translateY = transform . translationY -- Reflection ---------------------------------------------- -- | Construct a transformation which flips a diagram from left to -- right, i.e. sends the point (x,y) to (-x,y). reflectionX :: T2 reflectionX = scalingX (-1) -- | Flip a diagram from left to right, i.e. send the point (x,y) to -- (-x,y). reflectX :: (Transformable t, V t ~ R2) => t -> t reflectX = transform reflectionX -- | Construct a transformation which flips a diagram from top to -- bottom, i.e. sends the point (x,y) to (x,-y). reflectionY :: T2 reflectionY = scalingY (-1) -- | Flip a diagram from top to bottom, i.e. send the point (x,y) to -- (x,-y). reflectY :: (Transformable t, V t ~ R2) => t -> t reflectY = transform reflectionY -- | @reflectionAbout p v@ is a reflection in the line determined by -- the point @p@ and vector @v@. reflectionAbout :: P2 -> R2 -> T2 reflectionAbout p v = conjugate (rotation (-direction v :: Rad) <> translation (origin .-. p)) reflectionY -- | @reflectAbout p v@ reflects a diagram in the line determined by -- the point @p@ and the vector @v@. reflectAbout :: (Transformable t, V t ~ R2) => P2 -> R2 -> t -> t reflectAbout p v = transform (reflectionAbout p v) -- Shears -------------------------------------------------- -- | @shearingX d@ is the linear transformation which is the identity on -- y coordinates and sends @(0,1)@ to @(d,1)@. shearingX :: Double -> T2 shearingX d = fromLinear (over r2 (sh d) <-> over r2 (sh (-d))) (over r2 (sh' d) <-> over r2 (sh' (-d))) where sh k (x, y) = (x+k*y, y) sh' k = swap . sh k . swap swap (x,y) = (y,x) -- | @shearX d@ performs a shear in the x-direction which sends -- @(0,1)@ to @(d,1)@. shearX :: (Transformable t, V t ~ R2) => Double -> t -> t shearX = transform . shearingX -- | @shearingY d@ is the linear transformation which is the identity on -- x coordinates and sends @(1,0)@ to @(1,d)@. shearingY :: Double -> T2 shearingY d = fromLinear (over r2 (sh d) <-> over r2 (sh (-d))) (over r2 (sh' d) <-> over r2 (sh' (-d))) where sh k (x,y) = (x, y+k*x) sh' k = swap . sh k . swap swap (x,y) = (y,x) -- | @shearY d@ performs a shear in the y-direction which sends -- @(1,0)@ to @(1,d)@. shearY :: (Transformable t, V t ~ R2) => Double -> t -> t shearY = transform . shearingY -- Scale invariance ---------------------------------------- -- XXX what about freezing? Doesn't interact with ScaleInv the way it -- ought. -- | The @ScaleInv@ wrapper creates two-dimensional /scale-invariant/ -- objects. Intuitively, a scale-invariant object is affected by -- transformations like translations and rotations, but not by scales. -- -- However, this is problematic when it comes to /non-uniform/ -- scales (/e.g./ @scaleX 2 . scaleY 3@) since they can introduce a -- perceived rotational component. The prototypical example is an -- arrowhead on the end of a path, which should be scale-invariant. -- However, applying a non-uniform scale to the path but not the -- arrowhead would leave the arrowhead pointing in the wrong -- direction. -- -- Moreover, for objects whose local origin is not at the local -- origin of the parent diagram, any scale can result in a -- translational component as well. -- -- The solution is to also store a point (indicating the location, -- /i.e./ the local origin) and a unit vector (indicating the -- /direction/) along with a scale-invariant object. A -- transformation to be applied is decomposed into rotational and -- translational components as follows: -- -- * The transformation is applied to the direction vector, and the -- difference in angle between the original direction vector and its -- image under the transformation determines the rotational -- component. The rotation is applied with respect to the stored -- location, rather than the global origin. -- -- * The vector from the location to the image of the location under -- the transformation determines the translational component. data ScaleInv t = ScaleInv { unScaleInv :: t , scaleInvDir :: R2 , scaleInvLoc :: P2 } deriving (Show) -- | Create a scale-invariant object pointing in the given direction. scaleInv :: t -> R2 -> ScaleInv t scaleInv t d = ScaleInv t d origin type instance V (ScaleInv t) = R2 instance (V t ~ R2, HasOrigin t) => HasOrigin (ScaleInv t) where moveOriginTo p (ScaleInv t v l) = ScaleInv (moveOriginTo p t) v (moveOriginTo p l) instance (V t ~ R2, Transformable t) => Transformable (ScaleInv t) where transform tr (ScaleInv t v l) = ScaleInv (trans . rot $ t) (rot v) l' where angle :: Rad angle = direction (transform tr v) - direction v rot :: ( Transformable t, (V t ~ R2) ) => t -> t rot = rotateAbout l angle l' = transform tr l trans = translate (l' .-. l)