This work deals with trajectory optimization for a robotic sensor network sampling a spatio-temporal random field. We examine the optimal sampling problem of minimizing the maximum predictive variance of the estimator over the space of network trajectories. This is a high-dimensional, multimodal, nonsmooth optimization problem, known to be NP-hard even for static fields and discrete design spaces. Under an asymptotic regime of near-independence between distinct sample locations, we show that

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... solutions to a novel generalized disk-covering problem are solutions to the optimal sampling problem. This result effectively transforms the search for the optimal trajectories into a geometric optimization problem. Constrained versions of the latter are also of interest as they can accommodate trajectories that satisfy a maximum velocity restriction on the robots. We characterize the solution for the unconstrained and constrained versions of the geometric optimization problem as generalized multicircumcenter trajectories, and provide algorithms which enable the network to find them in a distributed fashion. Several simulations illustrate our results.