Documentation

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 0.45

 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)

Reciprocal of curvature.

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.

Reciprocal of squaredCurvature