This paper presents a novel approach to computing minimum-time paths based on a technique originally developed for use in geophysics. The technique is based on a finite difference scheme and is quite efficient in terms of both computing effort and storage. A particular strength of the technique is that it provides optimal paths to all locations in the field, thus being useful in situations where the goal is not known at the outset of the search. Details are presented on the basic technique and extensions are derived to include the integrations of resources (such as fuel store), and to provide flexibility in the objective. Two examples are given: minimum-time planning for vehicles under the influence of drift, and planning to minimise the risk of detection for a stealthy vehicle in the presence of threats.