This paper presents a path-planning control for an intermittently actuated oceanographic vehicle in a time-invariant flow field based on Lagrangian measurement data. The oceanographic focus is an idealized ocean-eddy system represented by two point vortices. We partition the model of the underlying flow into distinct invariant regions (in a co-rotating frame) and use this geometry to plan informative vehicle tours. Driving along the invariant-set boundaries increases the overall empirical observability of the flow field parameters. We validate the importance of invariant sets and set boundaries using the local unobservability index. To tour the boundaries of invariant sets, a novel control law drives the vehicle to a desired streamline value; another controller is based on the idea of inserting a virtual cylinder into the flow. Numerical experiments in the two-vortex system show the proposed Boundary-Touring Algorithm yields planned paths with high observability of the flow field parameters as compared to random drifter orbits.