Book about clothoids

Question

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.

Answer 1

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.