Abstract We want to compute geometric shortest paths in a collection of convex traversable polygons, also known as a navigation mesh. Simple to compute and easy to update, navigation meshes are widely used for pathfinding in computer games. When the mesh is static, shortest path problems can be solved exactly and very fast but only after a costly preprocessing step. When the mesh is dynamic, practitioners turn to online methods which typically compute only approximately shortest paths. In this work we present a new pathfinding algorithm which is compromise-free; i.e., it is simultaneously fast, online and optimal. Our method, Polyanya, extends and generalises Anya; a recent and related interval-based search technique developed for computing geometric shortest paths in grids. We show how that algorithm can be modified to support search over arbitrary sets of convex polygons and then evaluate its performance on a range of realistic and synthetic benchmark problems.