Computing the distance from a point to a helix and solving Kepler's equation

Computing the distance from a point to a helix is logically equivalent to solving Kepler's equation for elliptic motions. The “eccentricity” reflects the relation between the relative positions of the point and the helix, and the ratio of the curvature to the torsion of the helix. In particular, the “eccentricity” may have any non-negative value, which may exceed 1. Nevertheless, numerical analysis confirms that Newton's and Halley's methods converge to the solution from initial values provided by established algorithms from celestial mechanics.