ellipse_trig_free.py

# Trig free modification contributed by
# @adrianstephens and @JohannesMP

def solve(semi_major, semi_minor, p):  
    px = abs(p[0])
    py = abs(p[1])

    tx = 0.707
    ty = 0.707

    a = semi_major
    b = semi_minor

    for x in range(0, 3):
        x = a * tx
        y = b * ty

        ex = (a*a - b*b) * tx**3 / a
        ey = (b*b - a*a) * ty**3 / b

        rx = x - ex
        ry = y - ey

        qx = px - ex
        qy = py - ey

        r = math.hypot(rx, ry)
        q = math.hypot(qx, qy)

        tx = min(1, max(0, (qx * r / q + ex) / a))
        ty = min(1, max(0, (qy * r / q + ey) / b))
        t = math.hypot(tx, ty)
        tx /= t 
        ty /= t 

    return (math.copysign(a * tx, p[0]), math.copysign(b * ty, p[1]))