t-design curves and mobile sampling on the sphere

TL;DR

This paper introduces t-design curves on the sphere, which exactly integrate polynomials along their length, and proves their existence and optimality in various dimensions, extending classical spherical t-design concepts.

Contribution

It defines t-design curves on the sphere, constructs explicit low-degree examples, and proves the existence of asymptotically optimal t-design curves in multiple dimensions.

Findings

Explicit low-degree t-design curves constructed

Lower bounds on lengths of t-design curves derived

Existence of asymptotically optimal t-design curves proven

Abstract

In analogy to classical spherical t-design points, we introduce the concept of t-design curves on the sphere. This means that the line integral along a t-design curve integrates polynomials of degree t exactly. For low degrees we construct explicit examples. We also derive lower asymptotic bounds on the lengths of t-design curves. Our main results prove the existence of asymptotically optimal t-design curves in the Euclidean 2-sphere and the existence of t-design curves in the d-sphere.