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I'm looking for a good book (or a reliable website) about clothoids (aka Euler's spiral). I need a proper definition, the derivation of the curve and parametrization using Fresnel integrals. I only found this http://mathworld.wolfram.com/CornuSpiral.html but I need something more in-depth and I can't find Bernoulli's Opera. Any other suggestion? Thank you.

xyzt
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  • Can you be more specific about what you need to to know about Euler's spiral? I have several books on curves and spirals and it never warrants more than a page. The Wiki page has some interesting information about it. As for the Fresnel parameterization, this can be be expressed in a single closed-form equation in the complex plane. – Cye Waldman Aug 27 '17 at 17:24
  • @Cye Waldman I need a bibliography for my thesis (and I prefer not to use Wikipedia). Every book you could suggest about curves and spirals could be useful even if there are not too many pages about clothoids. I need its definition, some formulas (curvature, natural equation etc..) and how it is obtained. Thanks – xyzt Aug 29 '17 at 10:16

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Here is a list of books that I have with some discussion, albeit brief, of the Euler/Cornu spiral (or clothoid).

Frank Oliver et al., NIST Handbook of Mathematical Functions, Cambridge, 2010.

Eric Weisstein, CRC Encyclopedia of Mathematics, CRC Press, Chapman & Hall, 2003.

Philip Davis, Spirals: from Theodorus to Chaos, A K Peters, 1993.

J Dennis Lawrence, A Catalog of Special Plane Curves, Dover, 1972.

Davin von Seggern, CRC Standard Curves and Sufraces, CTC Press, 1993.

My favorite equation for the Cornu spiral spiral is the compact complex form that incorporates the two Fresnel integrals, i.e.,

$$ x\left( t \right) =C\left( t \right)=\int_{0}^{t}{\cos \left( \tfrac{\pi }{2}{{u}^{2}} \right)du} \\ y\left( t \right) =S\left( t \right)=\int_{0}^{t}{\sin \left( \tfrac{\pi }{2}{{u}^{2}} \right)du} \\ \\ \begin{align} z\left( t \right)&= C\left( t \right)+i\,S\left( t \right)=\int_{0}^{t}{{{e}^{i\tfrac{\pi }{2}{{u}^{2}}}}du} \\ & =\frac{1+i}{2}\text{erf}\left( \frac{1-i}{2}\sqrt{\pi }\cdot t \right) \end{align} $$

Finally, it's worth noting that the Cornu spiral is a special case of the more general polynomial spiral developed by

Dillen, F. (1990). "The Classification of Hypersurfaces of a Euclidean Space with Parallel Higher Order Fundamental Form," Mathematische Zeitschrift, 203: 635-643. (You should be able to find this article online.)

and also shown here.

Cye Waldman
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