Bézier Curve Calculator

def cubic_bezier(p0, p1, p2, p3, t):
    """Evaluate a cubic Bézier curve at parameter t (0..1)."""
    u = 1 - t
    return (
        u**3 * p0[0] + 3*u**2*t * p1[0] + 3*u*t**2 * p2[0] + t**3 * p3[0],
        u**3 * p0[1] + 3*u**2*t * p1[1] + 3*u*t**2 * p2[1] + t**3 * p3[1],
    )

def bezier_length(p0, p1, p2, p3, steps=200):
    """Approximate curve length via numerical integration."""
    length = 0.0
    prev = cubic_bezier(p0, p1, p2, p3, 0)
    for i in range(1, steps + 1):
        curr = cubic_bezier(p0, p1, p2, p3, i / steps)
        dx, dy = curr[0] - prev[0], curr[1] - prev[1]
        length += (dx**2 + dy**2) ** 0.5
        prev = curr
    return length

def bezier_tangent(p0, p1, p2, p3, t):
    """Calculate the tangent direction at parameter t."""
    u = 1 - t
    dx = 3*u**2*(p1[0]-p0[0]) + 6*u*t*(p2[0]-p1[0]) + 3*t**2*(p3[0]-p2[0])
    dy = 3*u**2*(p1[1]-p0[1]) + 6*u*t*(p2[1]-p1[1]) + 3*t**2*(p3[1]-p2[1])
    return (dx, dy)

# Example: S-curve
p0, p1, p2, p3 = (0, 0), (0.5, 1), (0.5, -1), (1, 0)
length = bezier_length(p0, p1, p2, p3)
print(f"Curve length: {length:.4f}")
print(f"Point at t=0.5: {cubic_bezier(p0, p1, p2, p3, 0.5)}")

# Sample 10 points
for i in range(11):
    t = i / 10
    pt = cubic_bezier(p0, p1, p2, p3, t)
    print(f"t={t:.1f}: ({pt[0]:.3f}, {pt[1]:.3f})")