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.