## Cornu Spiral

A plot in the Complex Plane of the points

 (1)

where and 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

 (2)

is plotted above.

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

 (3)

plotted below.

The Cesàro Equation for a Cornu spiral is , where is the Radius of Curvature and the Arc Length. The Torsion is .

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

 (4) (5)

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

 (6) (7) (8)

The Cesàro Equation is
 (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.

References

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.

© 1996-9 Eric W. Weisstein
1999-05-25