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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.

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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. 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, 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.

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The overview of approximation schemes for clothoids given in An approximation approach of the clothoid curve (2001) seems quite complete.

Because Taylor expansion series are not suitable for the calculation of the Fresnel integrals, Boersma [3], Cody [4] and Heald [5] proposed to use rational functions together with sine and cosine functions to approximate the Fresnel integrals. Meek and Walton [6] discussed in detail about how to use Cornu spirals as the transition curves between arcs, lines, or a line and an arc. Stoer [7] suggested least square interpolation with curves made up of Cornu spirals. Walton and Meek [8] and [9] discussed the construction of interpolation curves composed of Cornu spirals. We propose to approximate the clothoid curve defined in the interval $[0, \pi/2]$ and its offset by Bézier or B-spline curves.

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  • $\begingroup$ thanks for the pointer to the overview; unfortunately the article is not available not for free via the supplied link. $\endgroup$
    – Manfred Weis
    Apr 24, 2014 at 6:38
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I meanwhile found the following more recent articles, that are aimed at practical use:

Enrico Bertolazzi and Marco Frego: "Fast and accurate clothoid fitting", from Sept. 5th 2012 http://arxiv.org/pdf/1209.0910.pdf and,
David M. Smith: "Algorithm 911: Multiple-precision exponential integral and related functions", ACM Trans. Math. Softw. 37 (4), 46:1–46:16. http://doi.acm.org/10.1145/1916461.1916470

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