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Cornu Spiral


CornuSpiral

The Cornu spiral is a plot in the complex plane of the points

 B(t)=S(t)+iC(t),
(1)

where S(t) and C(t) 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.

CornuSlope

The quantities C(t)/S(t) and S(t)/C(t) are plotted above.

CornuNormalTangent

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

 m_T(t)=(S^'(t))/(C^'(t))=tan(1/2pit^2),
(2)

plotted below.

CornuTangentSlope

The Cesàro equation for a Cornu spiral is rho=c^2/s, where rho is the radius of curvature and s the arc length. The torsion is tau=0.

CornuSpirals

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

x(t)=aint_0^tsin((u^(n+1))/(n+1))du
(3)
=(at^(n+2))/((n+1)(n+2))_1F_2(1/2+1/(2(n+1));3/2,3/2+1/(2(n+1));-(t^(2(n+1)))/(4(n+1)^2))
(4)
y(t)=aint_0^tcos((u^(n+1))/(n+1))du
(5)
=at_1F_2(1/(2(n+1));1/2,1+1/(2(n+1));-(t^(2(n+1)))/(4(n+1)^2)),
(6)

where _1F_2(a;b,c;x) is a generalized hypergeometric function.

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

s(t)=at
(7)
kappa(t)=-(t^n)/a
(8)
phi(t)=-(t^(n+1))/(n+1).
(9)

The Cesàro equation is

 kappa=-(s^n)/(a^(n+1)).
(10)
CornuPolynomialSpirals

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 kappa=s, kappa=s^2, kappa=s^2-2.19, kappa=s^2-4, kappa=s^2+1, and kappa=5s^4-18s^2+5, respectively.


See also

Fresnel Integrals, Nielsen's Spiral

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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.

Cite this as:

Weisstein, Eric W. "Cornu Spiral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CornuSpiral.html

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