Inspired from this matlab implementation, wrote this python implementation of how to detect intersection of two curves.
from intersect import intersection
a, b = 1, 2
phi = np.linspace(3, 10, 100)
x1 = a*phi - b*np.sin(phi)
y1 = a - b*np.cos(phi)
x2 = phi
y2 = np.sin(phi)+2
x, y = intersection(x1, y1, x2, y2)
plt.plot(x1, y1, c="r")
plt.plot(x2, y2, c="g")
plt.plot(x, y, "*k")
plt.show()
To install
pip install intersect