Closest Points on a Circular Helix and Kepler’s Equation: Exact Counting, the Focal Surface, and the Principal-Branch Minimizer
Year: 2026
Type: article
Abstract: The closest point on a circular helix to a query point---a primitive of track fitting, computer-aided design, and robotics---reduces exactly to Kepler's equation \( u+e\sin u=M \), whose eccentricity \( e=a\rho/b^{2} \) can be arbitrarily large. We determine the complete solution structure of this problem. The cylinder law states that for \( rho < = b^{2}/a \) the squared distance is convex with a unique minimizer. The count law gives, for \( e>1 \), the exact number of stationary points \( N=2⌈{(M+c)/2\pi⌉ }-2⌊{(M-c)/2\pi}⌋-3 \) with \( c=\arccos(-1/e)+\sqrt{e^{2}-1} \); pairs of stationary points are born and annihilated on explicit spiral surfaces, which we identify with the focal surfaces of the helix, whose cuspidal edge is the evolute helix \( \rho=b^{2}/a \). The count grows like \( 2e/\pi \), so the often-quoted bound of three roots fails beyond \( e^{*}\approx4.6033 \). The principal-branch theorem shows that the global minimizer is always the unique root in the branch \( k_{0}=\mathrm{round}(M/2\pi) \), with the exact exception of two symmetric minimizers when \( M\equiv\pi\ (\mathrm{mod}\ 2\pi) \); the set of ties is the cut locus, terminating on the evolute helix. The minimizer is provably well-conditioned, \( |du^{*}/dM|< 1/(e-1) \), and conditioning degenerates only as \( O(\sqrt\delta) \) on the focal surfaces and as \( O(\delta^{1/3}) \) on the evolute. A hybrid bisection--Newton solver, a two-helix extension, and an exhaustive numerical verification complete the paper.
Source: Mathematics
DOI: 10.3390/math14203674
Author: Peilin Luo
Institution: Northeastern University
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,137
OpenAlex ID: W7221568017