t-Design Curves and Mobile Sampling on the Sphere

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.

Keywords

• 41A55
• 41A63
• 94A12
• 26B15