A spherical $t$-design curve is a curve on the $d$-dimensional sphere such that the corresponding line integral integrates polynomials of degree $t$ exactly. Spherical $t$-design curves can be used for mobile sampling and reconstruction of functions on the sphere.
(i) We derive lower asymptotic bounds for the length of $t$-design curves.
(ii) For the unit sphere in $\mathbb{R}^3$ and small degrees, we present examples of $t$-design curves with small $t$.
(iii) We prove the existence of asymptotically optimal $t$-design curves in the Euclidean $2$-sphere. This construction isbased on and uses the existence of $t$-design points verified by Bondarenko, Radchenko, and Viazovska (2013). For higher-dimensionalspheres we inductively prove the existence of $t$-design curves.
More generally, one can study the concept of t-design curves on a compact Riemannian manifold. This means that the line integral along a $t$-design curve integrates "polynomials" of degree $t$ exactly. For the $d$-dimensional tori, we construct $t$-design curves with asymptotically optimal length.
This is joint work with Martin Ehler and Clemens Karner, Univ. Vienna.