Closest Points on a Circular Helix and Kepler’s Equation: Exact Counting, the Focal Surface, and the Principal-Branch Minimizer

Source abstract

The closest-point problem for a circular helix reduces to Kepler’s equation u+esinu=M with geometric eccentricity e=aρ/b2 unrestricted above. Building on this known reduction and earlier half-turn localization, we derive an exact stationary-point count and a theorem selecting the global minimizer. For e≤1, the squared distance is strictly convex and has a unique strict minimum, including the degenerate boundary case. For e>1, the generic count is N=2(M+c)/2π−2(M−c)/2π−3, where c=arccos(−1/e)+e2−1. Separate indicators give the distinct and multiplicity counts on every fold, including simultaneous double roots. The count grows like 2e/π; five distinct stationary points become possible precisely when e>4.6033…. We express the classical normal-plane envelope, called the focal surface here, as explicit bifurcation sheets and determine the cut locus and its termination at the evolute helix. The principal-branch theorem identifies the nearest-turn root as the unique global minimizer with exactly two minimizers when M≡π(mod2π) and e>1. For fixed e>1, its absolute root condition number is bounded by 1/(e−1). Fold separation and cusp displacement scale as square and cube roots of the data offset; the corresponding condition numbers diverge with exponents −1/2 and −2/3. A bisection–Newton solver, an equal-pitch two-helix reduction, and reproducible numerical checks complement the analysis.