Fresnel Integral - 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 also expresses the unit tangent vector along the spiral, giving θ = t². Since t is the curve length, the curvature, can be expressed as:

And 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. That is, a vehicle following the spiral at constant speed will have a constant rate of angular acceleration.

Sections from Euler spirals are commonly incorporated into the shape of roller-coaster loops to make what are known as "clothoid loops".