Short spherical $t$-design curves
Provides the first exact optimality results for spherical t-design curves with t>1, advancing the theory of spherical designs for curves.
The authors prove a spectral lower bound on the arclength of spherical t-design curves, achieving sharpness for t=1 in all spheres and t=2 in odd-dimensional spheres, and construct near-optimal curves for even dimensions.
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.