Codes and Expansions (CodEx) Seminar

Martin Ehler (University of Vienna):
Spherical designs: from points to curves and beyond

In analogy to classical spherical \(t\)-design points, we study the concept of \(t\)-design curves on the sphere. While spherical design points can provide a model for static sensors distributed over the surface of the unit ball, the concept of spherical design curves accounts for mobile sampling, where a single sensor moves along the path of a curve on the sphere.

First, we provide several explicit examples of spherical t-design curves based on convex polytopes. We then derive lower bounds on the lengths of spherical t-design curves and prove the existence of asymptotically optimal t-design curves in the \(2\)-sphere that match this lower bound.

In practice it seems reasonable to use both, static sensors where available and in addition a mobile device. Therefore, we also propose the new notion of mixed designs that consist of points and a curve.

(This is joint work with Karlheinz Groechenig and Clemens Karner)