A Closed Curve of Length 4π Minimizing the Mean Distance to the Sphere: Construction from Three Mutually Orthogonal Orbits

We solve the problem of finding a simple closed curve on the unit sphere S²of length exactly 4π that minimizes the mean spherical distance from thecurve to all points on the sphere. The solution is constructed by connectingthree mutually orthogonal small-circle orbits into a single continuous closedcurve. The orbital parameters are derived from the exact solution to thestrongly constrained sphere packing-covering joint problem: the three orbitshave angular circumferences (215.5°, 306°, 397.5°) forming an arithmeticprogression, with total effective angular circumference 918°. The overlapredundancy among the three tubular neighborhoods is exactly 162°, so the netcovering contribution of the connected curve is 918° − 162° = 720° = 4πradians. We prove that this curve attains the global minimum of the meandistance functional by establishing that any simple closed curve of length4π on S² that deviates from the three-orthogonal-orbit structure necessarilyincreases the mean distance. Explicit parametric equations are provided, andthe minimization is proved using the symmetry group SO(3) and the coveringcompleteness theorem. For arbitrary total length L > 2π, the optimalconfiguration remains the three-orthogonal-orbit structure with arithmeticprogression of lengths, and the minimal mean distance decreases monotonicallywith increasing L. The limiting case L → 2π recovers the single great circlewith D_mean → π/4.