Short spherical $t$-design curves

Emily J. King; Dustin G. Mixon

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arXiv · 2607.15386

Short spherical $t$-design curves

Emily J. King·Dustin G. Mixon

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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 $d\to\infty$, and in $S^2$, we use numerical optimization and the calculus of variations to derive a candidate for the shortest $2$-design curve.

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BibTeX RIS

Emily J. King, Dustin G. Mixon. 2026-07-16. Short spherical $t$-design curves. https://arxiv.org/abs/2607.15386

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