Clothoid 

S(x) and C(x) The maximum of C(x) is about 0.977451424. If πt²/2 were used instead of t², then the image would be scaled vertically and horizontally (see below).

Fresnel integrals, S(x) and C(x), are two transcendental functions named after Augustin-Jean Fresnel that are used in optics. They arise in the description of near field Fresnel diffraction phenomena, and are defined through the following integral representations:

S(x)=\int_0^x \sin(t^2)\,dt,\quad C(x)=\int_0^x \cos(t^2)\,dt.

The simultaneous parametric plot of S(x) and C(x) is the Cornu spiral, or clothoid.

Contents

Definition

Normalised Fresnel integrals, S(x) and C(x). In these curves, the argument of the trigonometric function is πt2/2, as opposed to just t2 as above.

The Fresnel integrals admit the following power series expansions that converge for all x:

S(x)=\int_0^x \sin(t^2)\,dt=\sum_{n=0}^{\infin}(-1)^n\frac{x^{4n+3}}{(4n+3)(2n+1)!},
C(x)=\int_0^x \cos(t^2)\,dt=\sum_{n=0}^{\infin}(-1)^n\frac{x^{4n+1}}{(4n+1)(2n)!}.

Some authors, including Abramowitz and Stegun, (eqs 7.3.1 – 7.3.2) use \frac{\pi}{2}t^2 for the exponent of the integrals defining S(x) and C(x). To get the same functions, multiply the integral by \sqrt{\frac{2}{\pi}} and divide the argument x by the same factor.

Cornu spiral

Cornu spiral (x, y) = (C(t), S(t)). The spiral converges to the centre of the holes in the image as t tends to positive or negative infinity.

The Cornu spiral, also known as clothoid, is the curve generated by a parametric plot of S(t) against C(t). The Cornu spiral was created by Marie Alfred Cornu as a nomogram for diffraction computations in science and engineering.

Since

C'(t)2 + S'(t)2 = sin2(t2) + cos2(t2) = 1,

in this parametrization the tangent vector has unit length and t is the oriented arc length of the curve measured from the origin (0,0). Therefore, both spirals have infinite length.

It has the property that its curvature at any point is proportional to the distance along the curve, measured from the origin. This property makes it useful as a transition curve in highway and railway engineering, because a vehicle following the curve at constant speed will have a constant rate of angular acceleration. Sections from the clothoid spiral are commonly incorporated into the shape of roller-coaster loops to make what are known as "clothoid loops".

Properties

S(x)=\frac{\sqrt{\pi}}{4} \left( \sqrt{i}\,\operatorname{erf}(\sqrt{i}\,x) + \sqrt{-i}\,\operatorname{erf}(\sqrt{-i}\,x) \right)
C(x)=\frac{\sqrt{\pi}}{4} \left( \sqrt{-i}\,\operatorname{erf}(\sqrt{i}\,x) + \sqrt{i}\,\operatorname{erf}(\sqrt{-i}\,x) \right).
\int_{0}^{\infty} \cos t^2\,dt = \int_{0}^{\infty} \sin t^2\,dt = \frac{\sqrt{2\pi}}{4} = \sqrt{\frac{\pi}{8}}.

Evaluation

The sector contour used to calculate the limits of the Fresnel integrals

The limits of C and S as the argument tends to infinity can be found by the methods of complex analysis. This uses the contour integral of the function

e^{-\frac{1}{2}t^2}

around the boundary of the sector-shaped region in the complex plane formed by the positive x-axis, the half-line y = x, x ≥ 0, and the circle of radius R centered at the origin.

As R goes to infinity, the integral along the circular arc tends to 0, the integral along the real axis tends to the Gaussian integral

\int_{0}^{\infty} e^{-\frac{1}{2}t^2}dt = \sqrt{\frac {\pi}{2}},

and after routine transformations, the integral along the bisector of the first quadrant can be related to the limit of the Fresnel integrals.

See also

References