import numpy as np
import matplotlib.pyplot as plt

# Randomly generate an intersection point
intersection = np.random.uniform(-5, 5, 2)

# Randomly generate line angles through the intersection point
num_lines = 3
angles = np.random.uniform(0, np.pi, num_lines)
slopes = np.tan(angles)

# Define the lines: y = m(x - x0) + y0
def line(x, m, x0, y0):
    return m * (x - x0) + y0

# Plot the lines
x_vals = np.linspace(-10, 10, 400)
for m in slopes:
    plt.plot(x_vals, line(x_vals, m, intersection[0], intersection[1]), label=f'Line')

# Kaczmarz algorithm
def kaczmarz(A, b, x0, max_iter=20):
    x = x0
    history = [x0]
    for k in range(max_iter):
        i = k % A.shape[0]
        ai = A[i, :]
        bi = b[i]
        x = x + (bi - np.dot(ai, x)) / np.linalg.norm(ai)**2 * ai
        history.append(x)
    return np.array(history)

# Construct A and b from the lines
A = np.column_stack((-slopes, np.ones(num_lines)))
b = -slopes * intersection[0] + intersection[1]

# Initial guess
x0 = np.random.uniform(-5, 5, 2)

# Run Kaczmarz algorithm
history = kaczmarz(A, b, x0, max_iter=20)

# Plot the iterations
plt.scatter(history[:, 0], history[:, 1], color='red', zorder=5)
plt.plot(history[:, 0], history[:, 1], color='red', linestyle='--', label='Kaczmarz Iterations')

# Mark the intersection point
plt.scatter(intersection[0], intersection[1], color='green', marker='*', s=200, label='Solution', zorder=10)

# Set aspect ratio 1:1
plt.gca().set_aspect('equal', adjustable='box')

# Add labels, legend, and grid
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.grid()
plt.title('Kaczmarz Algorithm in $\mathbb{R}^2$')
plt.xlim(-10, 10)
plt.ylim(-10, 10)
plt.axhline(0, color='black', linewidth=0.5)
plt.axvline(0, color='black', linewidth=0.5)
plt.show()