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
where 
is a generalized hypergeometric function.
The arc length, curvature, and tangential angle of this curve are
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.
