Short spherical t-design curves

2026/07/16 by Emily J. King, Dustin G. Mixon
Computer Science · Mathematics · #math.CO #cs.NA #math.MG #math.NA

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Abstract

We study the minimum arclength of spherical t-design curves, i.e., closed rectifiable curves on S^d whose normalized arclength measure exactly integrates every polynomial of degree at most t. We prove an explicit spectral lower bound that is sharp for t=1 in all spheres and for t=2 in every odd-dimensional sphere, yielding the first exact optimality results for spherical t-design curves with t>1. For even-dimensional spheres, we construct 2-design curves whose lengths asymptotically match the lower bound as d→∞, and in S², we use numerical optimization and the calculus of variations to derive a candidate for the shortest 2-design curve.

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