PlanetPhysics/Fresnel integrals

For any real value of the argument , the Fresnel integrals and are defined as the integrals:

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

The functions C and S

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In optics, both of them express the intensity of diffracted light behind an illuminated edge.

Using the Taylor series expansions of cosine and sine, we get easily the expansions of the functions:

 

 

 
 

These converge for all complex values  , and thus define entire transcendental functions.

The Fresnel integrals at infinity have the finite value  

Clothoid

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The parametric presentation

 

represents a curve called clothoid .

Since the equations both define odd functions, the clothoid has symmetry about the origin.

The curve has the shape of a " " (see this diagram).

The arc length of the clothoid from the origin to the point , , is simply   Thus, the length of the whole curve to the point,   is infinite.

The curvature of the clothoid also is extremely simple,   i.e. proportional to the arc lenth; thus in the origin only the curvature is zero.

Conversely, if the curvature of a plane curve varies proportionally to the arc length, the curve is a clothoid.

This property of the curvature of clothoid is utilised in way and railway construction, since the form of the clothoid is very efficient when a straight portion of way must be bent to a turn, the zero curvature of the line can be continuously raised to the wished curvature.