import numpy as np
import matplotlib.pyplot as plt

# a, b, c, d in C
# ad - cb != 0
def moebius(z, a, b, c, d):
    return (a * z + b) / (c * z + d)

a = 1
b = 2
c = 3
d = 4

x = np.linspace(-10, 10, 100)
y = np.linspace(-10, 10, 100)
X, Y = np.meshgrid(x, y)
Z = X + 1j * Y

W = moebius(Z, a, b, c, d)

fig, ax = plt.subplots(subplot_kw={"projection": "3d"})
surf = ax.plot_surface(X, Y, W.real, cmap='inferno', linewidth=0)

ax.view_init(20, 235)
ax._axis3don = False
plt.tight_layout()
plt.show()
plt.savefig('moebius1234.png' ,bbox_inches='tight', transparent=True, dpi=1080)