diagrams-lib-0.7.1.1: Embedded domain-specific language for declarative graphics

Maintainer | diagrams-discuss@googlegroups.com
Safe Haskell | None

Diagrams.TwoD.Curvature

Description

Compute curvature for segments in two dimensions.

Synopsis

- curvature :: Segment Closed R2 -> Double -> PosInf Double
- radiusOfCurvature :: Segment Closed R2 -> Double -> PosInf Double
- squaredCurvature :: Segment Closed R2 -> Double -> PosInf Double
- squaredRadiusOfCurvature :: Segment Closed R2 -> Double -> PosInf Double

# Documentation

curvature Source

Arguments

:: Segment Closed R2 | Segment to measure on.
-> Double | Parameter to measure at.
-> PosInf Double | Result is a `PosInf` value where `PosInfty` represents infinite curvature or zero radius of curvature.

Curvature measures how curved the segment is at a point. One intuition for the concept is how much you would turn the wheel when driving a car along the curve. When the wheel is held straight there is zero curvature. When turning a corner to the left we will have positive curvature. When turning to the right we will have negative curvature.

Another way to measure this idea is to find the largest circle that we can push up against the curve and have it touch (locally) at exactly the point and not cross the curve. This is a tangent circle. The radius of that circle is the "Radius of Curvature" and it is the reciprocal of curvature. Note that if the circle is on the "left" of the curve, we have a positive radius, and if it is to the right we have a negative radius. Straight segments have an infinite radius which leads us to our representation. We result in a pair of numerator and denominator so we can include infinity and zero for both the radius and the curvature.

Lets consider the following curve:

The curve starts with positive curvature,

approaches zero curvature

then has negative curvature

     {-# LANGUAGE GADTs #-}

     import Diagrams.TwoD.Curvature
     import Data.Monoid.Inf
     import Diagrams.Coordinates

     segmentA = Cubic (12 & 0) (8 & 10) (OffsetClosed (20 & 8))

     curveA = lw 0.1 . stroke . fromSegments $ [segmentA]

     diagramA = pad 1.1 . centerXY $ curveA

     diagramPos = diagramWithRadius 0.2

     diagramZero = diagramWithRadius 0.5

     diagramNeg = diagramWithRadius 0.8

     diagramWithRadius t = pad 1.1 . centerXY
              $ curveA
             <> showCurvature segmentA t
              # withEnvelope (curveA :: D R2)
              # lw 0.05 # lc red

     showCurvature bez@(Cubic b c (OffsetClosed d)) t
       | v == 0    = mempty
       | otherwise = go (radiusOfCurvature bez t)
       where
         v@(x,y) = unr2 $ firstDerivative b c d t
         vp = (-y) & x

         firstDerivative b c d t = let tt = t*t in (3*(3*tt-4*t+1))*^b + (3*(2-3*t)*t)*^c + (3*tt)*^d

         go Infinity   = mempty
         go (Finite r) = (circle (abs r) # translate vpr
                      <> stroke (origin ~~ (origin .+^ vpr)))
                       # moveTo (origin .+^ atParam bez t)
           where
             vpr = r2 (normalized vp ^* r)

radiusOfCurvature Source

Arguments

:: Segment Closed R2 | Segment to measure on.
-> Double | Parameter to measure at.
-> PosInf Double | Result is a `PosInf` value where `PosInfty` represents infinite radius of curvature or zero curvature.

Reciprocal of `curvature`.

squaredCurvature :: Segment Closed R2 -> Double -> PosInf Double Source

With `squaredCurvature` we can compute values in spaces that do not support `sqrt` and it is just as useful for relative ordering of curvatures or looking for zeros.

squaredRadiusOfCurvature :: Segment Closed R2 -> Double -> PosInf Double Source

Reciprocal of `squaredCurvature`