E8.10The Euler spiral
As well as playing an important role in the description of diffraction effects in optics, the Fresnel integrals,
\[\begin{align*}
S(z) &= \int_0^z \sin\left(\frac{\pi t^2}{2}\right)\,\mathrm{d}t,\\
C(z) &= \int_0^z \cos\left(\frac{\pi t^2}{2}\right)\,\mathrm{d}t,\\
\end{align*}\]
find an application in the design of motorway junctions (freeway intersections). The curve described by the parametric equations \((x,y) = (S(t), C(t))\) is called a clothoid (or Euler spiral) and has the property that its curvature is proportional to the distance along the path of the curve. Hence, a vehicle travelling at constant speed will experience a constant rate of angular acceleration as it travels around the curve — this means that the driver can turn the steering wheel at a constant rate which makes the junction safer.
The following code plots the Euler spiral for \(-10 \le t \le 10\).
eulerspiral
import numpy as np
from scipy.special import fresnel
import matplotlib.pyplot as plt
t = np.linspace(-10, 10, 1000)
plt.plot(*fresnel(t), c="k")
plt.show()
