Cornu Spiral
A plot in the Complex Plane of the points
 | (1) |
and 
are the Fresnel Integrals. The Cornu spiral is also known as the Clothoid orEuler's Spiral. A Cornu spiral describes diffraction from the edge of a half-plane.
The Slope of the Cornu spiral
 | (2) |
The Slope of the curve's Tangent Vector (above right figure) is
 | (3) |
The Cesàro Equation for a Cornu spiral is 
, where 
is the Radius ofCurvature 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.
See also Fresnel Integrals, Nielsen's 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.
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