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Cornu Spiral

\begin{figure}\begin{center}\BoxedEPSF{cornu_spiral.epsf scaled 600}\end{center}\end{figure}

A plot in the Complex Plane of the points

B(z) = C(t)+iS(t)=\int_0^t e^{i\pi x^2/2}\,dx,
\end{displaymath} (1)

where $C(z)$ and $S(z)$ are the Fresnel Integrals. The Cornu spiral is also known as the Clothoid or Euler's Spiral. A Cornu spiral describes diffraction from the edge of a half-plane.


The Slope of the Cornu spiral

m(t)={S(t)\over C(t)}
\end{displaymath} (2)

is plotted above.


The Slope of the curve's Tangent Vector (above right figure) is

m_T(t)={S'(t)\over C'(t)}=\tan({\textstyle{1\over 2}}\pi t^2),
\end{displaymath} (3)

plotted below.


The Cesàro Equation for a Cornu spiral is $\rho=c^2/s$, where $\rho$ is the Radius of Curvature and $s$ the Arc Length. The Torsion is $\tau=0$.


Gray (1993) defines a generalization of the Cornu spiral given by parametric equations

$\displaystyle x(t)$ $\textstyle =$ $\displaystyle a\int_0^t \sin\left({u^{n+1}\over n+1}\right)\,du$ (4)
$\displaystyle y(t)$ $\textstyle =$ $\displaystyle a\int_0^t \cos\left({u^{n+1}\over n+1}\right)\,du.$ (5)

\begin{figure}\begin{center}\BoxedEPSF{CornuSpiralInfo.epsf scaled 700}\end{center}\end{figure}

The Arc Length, Curvature, and Tangential Angle of this curve are

$\displaystyle s(t)$ $\textstyle =$ $\displaystyle at$ (6)
$\displaystyle \kappa(t)$ $\textstyle =$ $\displaystyle -{t^n\over a}$ (7)
$\displaystyle \phi(t)$ $\textstyle =$ $\displaystyle -{t^{n+1}\over n+1}.$ (8)

The Cesàro Equation is
\kappa=-{a\over s^n}.
\end{displaymath} (9)

Dillen (1990) describes a class of ``polynomial spirals'' for which the Curvature is a polynomial function of the Arc Length. These spirals are a further generalization of the Cornu spiral.

See also Fresnel Integrals, Nielsen's Spiral


Dillen, F. ``The Classification of Hypersurfaces of a Euclidean Space with Parallel Higher Fundamental Form.'' Math. Z. 203, 635-643, 1990.

Gray, A. ``Clothoids.'' §3.6 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 50-52, 1993.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 190-191, 1972.

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© 1996-9 Eric W. Weisstein