{-# LANGUAGE TypeFamilies #-} ----------------------------------------------------------------------------- -- | -- Module : Diagrams.TwoD.Arc -- Copyright : (c) 2011 diagrams-lib team (see LICENSE) -- License : BSD-style (see LICENSE) -- Maintainer : diagrams-discuss@googlegroups.com -- -- Two-dimensional arcs, approximated by cubic bezier curves. -- ----------------------------------------------------------------------------- module Diagrams.TwoD.Arc ( arc , bezierFromSweep ) where import Graphics.Rendering.Diagrams import Diagrams.TwoD.Types import Diagrams.TwoD.Transform import Diagrams.TwoD.Util import Diagrams.Path import Diagrams.Segment import Data.VectorSpace((^-^)) -- For details of this approximation see: -- http://www.tinaja.com/glib/bezcirc2.pdf -- | @bezierFromSweepQ1 s@ constructs a 'Cubic' segment that starts in -- the positive y direction and sweeps counterclockwise through @s@ -- radians. The approximation is only valid for angles in the first -- quadrant. bezierFromSweepQ1 :: Rad -> Segment R2 bezierFromSweepQ1 s = fmap (^-^ v) . rotate (s/2) $ Cubic p2 p1 p0 where p0@(x,y) = rotate (s/2) v p1 = ((4-x)/3, (1-x)*(3-x)/(3*y)) p2 = reflectY p1 v = (1,0) -- | @bezierFromSweep s@ constructs a series of 'Cubic' segments that -- start in the positive y direction and sweep counter clockwise -- through @s@ radians. If @s@ is negative, it will start in the -- negative y direction and sweep clockwise. When @s@ is less than -- 0.0001 the empty list results. If the sweep is greater than two pi -- then it is truncated to two pi. bezierFromSweep :: Rad -> [Segment R2] bezierFromSweep s | s > tau = bezierFromSweep tau | s < 0 = fmap reflectY . bezierFromSweep $ (-s) | s < tau/4 = [bezierFromSweepQ1 s] | s < 0.0001 = [] | otherwise = bezierFromSweepQ1 (tau/4) : map (rotateBy (1/4)) (bezierFromSweep (max (s - tau/4) 0)) {- ~~~~ Note [segment spacing] There are a few obvious options for segment spacing: A. Evenly space segments each with sweep less than or equal to half pi. This has the benefit of a better approximation (at least I think it is better). B. Use as much of the sweep in half pi sized segments and one for the remainder. This potentially gives more opportunities for consistency (though not as much as option C) as the error in approximation would more often match the error from another arc in the diagram. C. Like option B but fixing the orientation and having a remnant at the beginning and the end. Option B is implemented and this note is for posterity if anyone comes across a situation with large enough arcs that they can actually see the approximation error. -} arcT :: Angle a => a -> a -> Trail R2 arcT start end = Trail bs (sweep >= tau) where sweep = convertAngle $ end - start bs = map (rotate start) . bezierFromSweep $ sweep -- | Given a start angle @s@ and an end angle @e@, @'arc' s e@ is the -- path of a radius one arc counterclockwise between the two angles. arc :: (Angle a, PathLike p, V p ~ R2) => a -> a -> p arc start end = setStart (rotate start $ P unitX) . pathLikeFromTrail $ arcT start end