Fast cubic Bézier curve lenght calculation library and benchmark.
We test 3 methods:
- The easiest one - line interpolation: We split the curve into n points, and return the sum of the distances between points.
- The fastest one - midpoint quadratic interpolation: We interpolate the cubic curve with quadratic one and calculate it's length from integral.
- The most precise - adaptive quadratic interpolation: We interpolate segments of the cubic curve with multiple quadratic curves and return the sum of their lengths calculated with integral.
Test sample:
Short railway track segment, length: ca 33m. For the line interpolation the precision was set to 0.05 (5cm). For integral interpolations the presision is 0.001 (1mm).
About test:
The tracks' acrs are almost ideally interpolated with quadratic Bézier curves and that's why we can't see any difference between midpoint and adaptive interpolations.
The advantage of adaptive interpolation will be visible with more complex Bézier curves.
The defect of the interpolation was not calculated. Feel free to add the calculations yourself. However - playing with imposed precission for adaptive algorithm I've noticed no segmentation triggered even for values smaller than 1ppm.
The length calculated with integral is precise. The precision of calculation in this example cannot be increased through iteration and the code itself need no iterations (except adaptive algorithm and more complex curves).
If you want to use the code with different vector libraries or number types - just change apropriate types in code. The vector class I wrote here is minimalistic, I could use generic type instead of double, but it would probably result a little slower code.
License and legal stuff:
Free / Open Source, all the hard work done by Adrian Colomitchi and Dave Eberly, I only made a C# library and benchmark. For more details and theory visit: http://caffeineowl.com/graphics/2d/vectorial/ http://www.gamedev.net/topic/551455-length-of-a-generalized-quadratic-bezier-curve-in-3d/ http://www.geometrictools.com/index.html