import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import brentq
# Parameters corresponds to the given theta and b but trying different t.
b=1.0
theta = np.pi/4
t=np.linspace(0, 2*np.pi, 100)

# Plotting the ball B2(0) of radius 2
fig, ax = plt.subplots(figsize=(10, 8))
circle = plt.Circle((0, 0), 2, color='blue', fill=False, label='Ball B2(0)')
ax.add_artist(circle)

# Parametrization of p for different values of k
k_values = [0.1, 0.5, 1, 2]  

# Plot parametrization of p for different values of k on the ball
for k in k_values:
    p_x = (1/k) * (np.sin(theta) + np.cos(t))
    p_y = b + (1/k) * (-np.cos(theta) + np.sin(t))
    ax.plot(p_x, p_y, label=f'k={k}')

# Plot the points (±2, 0)
ax.plot([2, -2], [0, 0], 'go', label='Points (±2, 0)')

# Setting the same scale for both axes and other plot settings
ax.set_xlim([-3, 3])
ax.set_ylim([-3, 3])
ax.set_aspect('equal', 'box')
ax.legend()
ax.grid(True)
ax.set_title('Parametrization of p for different k values and Arc through (±2, 0) on Ball B2(0)')
plt.show()




##################Plot the dynamical version of the parametrization of p for different k values on the ball B2(0) and the arc through (±2, 0)############################################

import os
import numpy as np
import matplotlib.pyplot as plt
import imageio
from scipy.optimize import brentq
os.makedirs('tmp', exist_ok=True)

# Define parametrization functions
def p_x(t, k, theta):
    return (1/k) * (np.sin(theta) + np.cos(t))

def p_y(t, k, theta, b):
    return b + (1/k) * (-np.cos(theta) + np.sin(t))

# Define the distance function from the origin
def distance_from_origin(t, k, theta, b):
    return np.sqrt(p_x(t, k, theta)**2 + p_y(t, k, theta, b)**2) - 2

# Check for sign changes in the distance function across the interval [0, 2pi]
def find_intersections(k, theta, b):
    t_values = np.linspace(0, 2 * np.pi, 1000)
    distances = distance_from_origin(t_values, k, theta, b)
    sign_changes = np.where(np.diff(np.sign(distances)))[0]  # Indices where sign changes
    intersections = []
    for idx in sign_changes:
        try:
            t_intersection = brentq(distance_from_origin, t_values[idx], t_values[idx + 1], args=(k, theta, b))
            intersections.append(t_intersection)
        except ValueError:
            pass 
    return intersections



# Function to plot parametrization of p for a given k on the ball B2(0)
def plot_for_k(k, filename, b, theta):
    intersections = find_intersections(k, theta, b)
    fig, ax = plt.subplots(figsize=(10, 8))
    circle = plt.Circle((0, 0), 2, color='blue', fill=False)
    ax.add_artist(circle)
    t = np.linspace(0, 2 * np.pi, 100)
    
    ax.plot(p_x(t, k, theta), p_y(t, k, theta, b), label=f'k={k}')

    # Points (±2, 0)
    ax.plot([2, -2], [0, 0], 'go')

    for t_int in intersections:
        P_int = (p_x(t_int, k, theta), p_y(t_int, k, theta, b))
        ax.plot(P_int[0], P_int[1], 'ro')

    # Line for k=0.05
    line_x = np.linspace(-3, 3, 100)
    line_y = np.tan(theta) * (line_x - 0) + b
    ax.plot(line_x, line_y, 'r--', label='Line for k=0.05')
        
    # Calculate and plot tangent at (0,b) for k
    # Placeholder tangent, replace with actual calculation
    tangent_y = np.tan(theta) * (line_x - 0) + b
    ax.plot(line_x, tangent_y, 'g--', label=f'Tangent at (0,{b}) for k={k}')

    ax.set_xlim([-3, 3])
    ax.set_ylim([-3, 3])
    ax.set_aspect('equal', 'box')
    ax.legend()
    plt.close(fig)
    fig.savefig(filename)
    return filename

def arc_equation(t, k, theta, b):
    p_x = (1/k) * (np.sin(theta) + np.cos(t))
    p_y = b + (1/k) * (-np.cos(theta) + np.sin(t))
    return p_x, p_y


# Generate plots for a series of k values and create a GIF
theta = np.pi/4 #0
b=1
k_values = list(np.linspace(0.05, 0.86, 20) )+ list(np.linspace(0.86, 0.87, 50) )+list(np.linspace(0.87, 4, 30))
filenames = []
for k in k_values:
    filename = f'tmp/plot_k_{k:.4f}.png'
    plot_for_k(k, filename,b, theta)
    filenames.append(filename)

# Create a GIF from the saved images
with imageio.get_writer('parameterized_theta=0.gif', mode='I', duration=0.5) as writer:
    for filename in filenames:
        image = imageio.imread(filename)
        writer.append_data(image)