Python code to test cosine weighted hemispherical 2d vectors

CosHemiVec.py

import matplotlib.pyplot as plt
import math
import random

numPoints = 10000
transparency = 0.01

def Normalize(x):
    len = math.sqrt(x[0] * x[0] + x[1] * x[1])
    return [x[0] / len, x[1] / len]

def PointOnCircle():
    angle = math.pi * 2.0 * random.random()
    radius = 1.0
    return [math.cos(angle) * radius, math.sin(angle) * radius]

def PointInCircle():
    angle = math.pi * 2.0 * random.random()
    radius = math.sqrt(random.random())
    return [math.cos(angle) * radius, math.sin(angle) * radius]

fig, ax = plt.subplots(1, 2, figsize=(12, 6), sharex=True, sharey=True)

ax[0].set_title("In Circle")
ax[1].set_title("On Circle")

for i in range(2):
    ax[i].set_xlim(-1, 1)
    ax[i].set_ylim(-0.5, 1.5)
    ax[i].plot([-2, 2], [0, 0], 'g-')

for i in range(numPoints):

    p = PointOnCircle()
    p[1] = p[1] + 1.0
    p = Normalize(p)
    ax[0].plot([0, p[0]], [0, p[1]], 'b-', alpha=transparency)
   
    p = PointInCircle()
    p[1] = p[1] + 1.0
    p = Normalize(p)
    ax[1].plot([0, p[0]], [0, p[1]], 'b-', alpha=transparency)

fig.savefig("out.png", bbox_inches='tight')