Inspired by https://www.youtube.com/watch?v=r6sGWTCMz2k

fourier.py

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation

fig, ax = plt.subplots()
ax.set_aspect("equal", adjustable="box")

(path,) = ax.plot([], [], marker="o", color="r", linewidth=1, markersize=0.2)
centres = ax.scatter([], [], marker="o", color="b", linewidth=0.1)


def square(t):
    p = t / (2 * np.pi)  # normalise to p in (0, 1)

    # Draw square between (0, 0), (1, 0), (0, 1), (1, 1)
    relative = 4 * p
    if relative < 1:
        result = relative + 1j
    elif relative < 2:
        result = 1 + (2 - relative) * 1j
    elif relative < 3:
        result = 3 - relative
    else:
        result = (relative - 3) * 1j

    # Scale and translate
    return result * 2 - (1 + 1j)


interval = np.linspace(0, 2 * np.pi, 200)


def calculate_fourier_coefficints(interval, function, N=30):
    """Calculates `N` (complex) fourier coefficients for a function"""
    coefficients = []
    function = np.vectorize(function)

    for n in range(-N, N):
        res = function(interval) * np.exp(-n * 1j * interval)
        cn = np.trapz(res) / (2 * np.pi)
        coefficients.append(cn)
    return coefficients


coefficients = calculate_fourier_coefficints(interval, square)

# Sort by size. Larger radiuses are drawn first
sorted_coeficients = sorted(
    list(enumerate(coefficients, start=-(len(coefficients) // 2))), key=lambda x: np.abs(x[1]), reverse=True
)

circles = [plt.Circle((0, 0), np.abs(c), color="g", fill=False) for (_, c) in sorted_coeficients]
for circle in circles:
    ax.add_patch(circle)

x_data, y_data = [], []


def update(t):
    x, y = 0, 0

    # v = my_function(t)
    # x, y = np.real(v), np.imag(v)

    centeres = []
    centeres.append((x, y))

    for i, (omega, c) in enumerate(sorted_coeficients):
        circles[i].set_center((x, y))

        p = c * np.exp(omega * 1j * t)
        x += np.real(p)
        y += np.imag(p)
        centeres.append((x, y))

    centres.set_offsets(centeres)

    # Final result of summation
    x_data.append(x)
    y_data.append(y)
    path.set_data(x_data, y_data)

    return [path, centres, *circles]


vp = 50
ax.set_xlim(-vp, vp)
ax.set_ylim(-vp, vp)

animation = FuncAnimation(fig, update, frames=interval, interval=50, blit=True)

plt.get_current_fig_manager().window.showMaximized()
plt.show()

# paused = False


# def toggle_pause(_):
#     global paused
#     if paused:
#         animation.resume()
#     else:
#         animation.pause()
#     paused = not paused

# from matplotlib.widgets import Button
# axes = plt.axes([0, 0, 0.2, 0.2])
# Button(axes, "Play/Pause", color="yellow")
# bnext.on_clicked(toggle_pause)