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.
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cornu spiral
