Real-time situated agents, including characters in real-time computer games, often do not know the terrain in advance but automatically observe it within a certain range around them. They have to interleave planning with movement to make planning tractable when moving autonomously to user-specified coordinates. Planning faces real-time requirements since it is important that the agents be responsive to the commands of the users and move smoothly. In this paper, we compare two fast search methods for this task that speed up planning in different ways, namely real-time heuristic search (LRTA*) and incremental heuristic search (D* Lite), resulting in the first comparison of real-time and incremental heuristic search in the literature. We characterize when to choose which search method, depending on the kind of terrain and the planning objective.