Lissajous Curve Generator

import math

def lissajous_points(freq_a: float, freq_b: float, delta: float,
                     amplitude: float = 1.0, num_points: int = 1000) -> list[tuple[float, float]]:
    """Generate Lissajous curve points.
    x(t) = A * sin(a*t + delta)
    y(t) = A * sin(b*t)
    """
    points = []
    for i in range(num_points):
        t = 2 * math.pi * i / num_points
        x = amplitude * math.sin(freq_a * t + delta)
        y = amplitude * math.sin(freq_b * t)
        points.append((x, y))
    return points

# Common interesting ratios
ratios = [
    (1, 1, math.pi / 4, "Ellipse (a:b=1:1)"),
    (1, 2, math.pi / 4, "Figure-8 (a:b=1:2)"),
    (2, 3, math.pi / 4, "Bow-tie (a:b=2:3)"),
    (3, 4, math.pi / 4, "Complex knot (a:b=3:4)"),
    (5, 4, 0,           "Star pattern (a:b=5:4)"),
]

for a, b, delta, label in ratios:
    pts = lissajous_points(a, b, delta)
    x_vals = [p[0] for p in pts]
    y_vals = [p[1] for p in pts]
    print(f"{label}: x range [{min(x_vals):.3f}, {max(x_vals):.3f}]")

# Number of lobes = |a - b| or related to ratio
# Closed curve when a/b is rational
def is_closed(a: float, b: float) -> bool:
    from math import gcd
    if isinstance(a, int) and isinstance(b, int):
        g = gcd(a, b)
        return True  # rational ratio
    return False

print(f"\na=3, b=4 closed curve: {is_closed(3, 4)}")