Cornu Spiral

A plot in the complex plane of the points

(1)

where  and  are the Fresnel integrals (von Seggern 2007, p. 210; Gray 1997, p. 65). The Cornu spiral is also known as the clothoid or Euler's spiral. It was probably first studied by Johann Bernoulli around 1696 (Bernoulli 1967, pp. 1084-1086). A Cornu spiral describes diffraction from the edge of a half-plane.

The quantities  and  are plotted above.

The slope of the curve's tangent vector (above right figure) is

(2)

plotted below.

The Cesàro equation for a Cornu spiral is , where  is the radius of curvature and  the arc length. The torsion is .

Gray (1997) defines a generalization of the Cornu spiral given by parametric equations

			(3)
			(4)
			(5)
			(6)

where  is a generalized hypergeometric function.

The arc length, curvature, and tangential angle of this curve are

			(7)
			(8)
			(9)

The Cesàro equation is

(10)

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. The curves plotted above correspond to , , , , , and , respectively.

REFERENCES:

Bernoulli, J. Opera, Tomus Secundus. Brussels, Belgium: Culture er Civilisation, 1967.

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

Gray, A. "Clothoids." §3.7 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 64-66, 1997.

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

von Seggern, D. CRC Standard Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 2007.