Clothoids

A clothoid is a curve \((x(s),y(x))\) is described by the parametric equations:

• \[ x(s)=\int_0^s \cos\left(\frac{1}{2}\kappa'\tau^2+\kappa_0\tau+\vartheta_0\right)\,\mathrm{d}\tau \]
• \[ y(s)=\int_0^s \sin\left(\frac{1}{2}\kappa'\tau^2+\kappa_0\tau+\vartheta_0\right)\,\mathrm{d}\tau \]

when \(\kappa'=0\) the clothoids reduce to a circle arc and when \(\kappa'=\kappa_0=0\) the clothoids reduce to a straight segment.

This library implements algorithms for \(G^1\) and \(G^2\) fitting with clothoids, spline of clothoids, circle arc and biarc.

The implementation of the algorithms on clothoids, splines of clothoids, arc, bi-arc splines of biarcs are described in the works:

• E.Bertolazzi, M.Frego, \(G^1\) fitting with clothoids, Mathematical Methods in the Applied Sciences, 2015, https://doi.org/10.1002/mma.3114

• E.Bertolazzi, M.Frego, Interpolating clothoid splines with curvature continuity, Mathematical Methods in the Applied Sciences, 2018, https://doi.org/10.1002/mma.4700

• E.Bertolazzi, M.Frego, On the G2 Hermite interpolation problem with clothoids, Journal of Computational and Applied Mathematics, 2018, https://doi.org/10.1016/j.cam.2018.03.029

• E.Bertolazzi, M.Frego, A Note on Robust Biarc Computation, Computer-Aided Design And Applications, 2019, https://doi.org/10.14733/cadaps.2019.822-835

• E.Bertolazzi, M.Frego, Point-Clothoid Distance and Projection Computation, SIAM Journal on Scientific Computing, 2019, https://doi.org/10.1137/18M1200439

• E.Bertolazzi, M.Frego, Francesco Biral, Interpolating splines of biarcs from a sequence of planar points, Computer-Aided Design And Applications, 2020, https://doi.org/10.14733/cadaps.2021.66-85

• E.Bertolazzi, P.Bevilacqua, M.Frego, Efficient intersection between splines of clothoids, Mathematics and Computers in Simulation, 2020, https://doi.org/10.1016/j.matcom.2019.10.001

• E.Bertolazzi, M.Frego, Francesco Biral, Interpolating splines of biarcs from a sequence of planar points, Computer-Aided Design And Applications, 2020, https://doi.org/10.14733/cadaps.2021.66-85

• M.Frego, Closed form parametrisation of 3D clothoids by arclength with both linear varying curvature and torsion, Applied Mathematics and Computation, 2022, https://doi.org/10.1016/j.amc.2021.126907

The library contains the following objects:

• Segment

• CircleArc

• Clothoids

• BiArc

• spline of

  • Segment

  • CircleArc

  • Clothoids (with \(G^1\) and \(G^2\) continuity)

  • BiArc

• Triangles

• BBox (bounding box)

and fast algorithms involving the objects, in particular:

• evaluation

• intersection (between objects)

• point-object distance

Library is written in C++11 with a MATLAB mex interface. Thus can be used in fast compiled application or in MATLAB scripts.

To compile the C++11 library the easy way require cmake and rake

ruby setup.rb

then

rake

for more details see: online documentation at http://ebertolazzi.github.io/Clothoids/