Background of my question is, that I need to calculate Clothoids and I found an AMS article "*Chebyhev Approximations for Fresnel Integrals*" by W.J. Cody from 1968 (<http://www.ams.org/journals/mcom/1968-22-102/S0025-5718-68-99871-2/S0025-5718-68-99871-2.pdf>). 


**Question:** is something better available/possible, especially due to advances in the interpolation of functions (cf e.g. Floater and Hormann: "*Barycentric rational interpolation with no poles and high rates of approximation (2006)*" <http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.115.9578>)? 


The interpretation of what is "better" is intentionally left open.



State of the Art in Approximating Fresnel Integrals
If you aim to compute clothoids to interpolate two given points with assigned tangents, you can have a look at the paper cited above by Bertolazzi and Frego. The arxiv version is updated with a published paper [G1 fitting with clothoids](http://onlinelibrary.wiley.com/doi/10.1002/mma.3114/abstract).
That paper is good because there is a comparison of the state of the art algorithms for clothoid fitting, there is a free matlab implementation [Matlab implementation](http://www.mathworks.com/matlabcentral/fileexchange/42113-g1-fitting-with-clothoids), and a discussion of the computation of the Generalized Fresnel Integrals, i.e. the moments. There are also proofs of accuracy and convergence with interesting counterexamples where other methods fail. The authors introduce also the quasi G2-interpolation, that is you can fix position and tangent and you get an interpolation with continuous curvature, while the G1 is only piecewise continuous.


