Named after Augustin-Jean Fresnel Fresnel integrals are:

\[x(t) = C(t) = \int_{0}^{t} cos(\dfrac{\pi}{2}s^2) ds\] \[y(t) = S(t) = \int_{0}^{t} sin(\dfrac{\pi}{2}s^2) ds\]

You can read more about them here.

%matplotlib inline
from scipy.special import fresnel
from scipy import linspace
import matplotlib.pyplot as plt

t = linspace(0, 5, 1000)
y, x = fresnel(t)
plt.figure(figsize=(10,10))
plt.plot(t, x, t, y)
plt.show()

Fresnel Integral

Euler spiral

The parametric plot of these functions is called the Euler spiral.

t = linspace(0, 10, 1000)
y, x = fresnel(t)
plt.figure(figsize=(10,10))
plt.plot(x, y)
plt.show()

Half Euler Spiral

t = linspace(-10, 10, 1000)
y, x = fresnel(t)
plt.figure(figsize=(10,10))
plt.plot(x, y)
plt.show()

Euler Spiral