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Concept
Fresnel integral
Summary
The Fresnel integrals S(x) and C(x) are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (erf). They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations:
The simultaneous parametric plot of S(x) and C(x) is the Euler spiral (also known as the Cornu spiral or clothoid).
The Fresnel integrals admit the following power series expansions that converge for all x:
Some widely used tables use π/2t2 instead of t2 for the argument of the integrals defining S(x) and C(x). This changes their limits at infinity from 1/2· to 1/2 and the arc length for the first spiral turn from to 2 (at t = 2). These alternative functions are usually known as normalized Fresnel integrals.
Euler spiral
The Euler spiral, also known as Cornu spiral or 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.
From the definitions of Fresnel integrals, the infinitesimals dx and dy are thus:
Thus the length of the spiral measured from the origin can be expressed as
That is, the parameter t is the curve length measured from the origin (0, 0), and the Euler spiral has infinite length. The vector (cos(t2), sin(t2)) also expresses the unit tangent vector along the spiral, giving θ = t2. Since t is the curve length, the curvature κ can be expressed as
Thus the rate of change of curvature with respect to the curve length is
An Euler spiral has the property that its curvature at any point is proportional to the distance along the spiral, measured from the origin. This property makes it useful as a transition curve in highway and railway engineering: if a vehicle follows the spiral at unit speed, the parameter t in the above derivatives also represents the time. Consequently, a vehicle following the spiral at constant speed will have a constant rate of angular acceleration.
Official source
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