Plot a discrete z-grid with frequency and damping

zgrid.py

'''
Author: Victor Maryama
Adapted from 
https://tex.stackexchange.com/questions/373314/how-to-draw-this-z-grid-graph-of-constant-damping-factors-and-natural-frequencie?noredirect=1&lq=1
'''

from numpy import pi, linspace
from scipy import sqrt, exp, cos, sin
import matplotlib.pyplot as plt

def zgrid():
    for zeta in linspace(0, 0.9,10):
        factor = zeta/sqrt(1-zeta**2)
        x = linspace(0, sqrt(1-zeta**2),200)
        ang = pi*x
        mag = exp(-pi*factor*x)
        xret = mag*cos(ang)
        yret = mag*sin(ang)
        plt.plot(xret,yret, 'k:', lw=1)
        xret = mag*cos(-ang)
        yret = mag*sin(-ang)    
        plt.plot(xret,yret,'k:', lw=1)
        an_i = int(len(xret)/2.5)
        an_x = xret[an_i]
        an_y = yret[an_i]
        plt.annotate(str(zeta), xy=(an_x, an_y), xytext=(an_x, an_y), size=7)

    for a in linspace(0,1,10):
        x = linspace(-pi/2,pi/2,200)
        ang = pi*a*sin(x)
        mag = exp(-pi*a*cos(x))
        xret = mag*cos(ang)
        yret = mag*sin(ang)
        plt.plot(xret,yret,'k:', lw=1)    
        an_i = -1 #int(len(xret)/2.5)
        an_x = xret[an_i]
        an_y = yret[an_i]
        num = '{:1.1f}'.format(a)
        plt.annotate("$\\frac{"+num+"\pi}{T}$", xy=(an_x, an_y), xytext=(an_x, an_y), size=8)