Feature-driven analysis forms the basis of many shape processing tasks, where detected feature points are characterized by local shape descriptors. Such descriptors have so far been defined to capture regions of interest centred at individual points. Using such regions to compare feature points can be problematic when performing partial shape matching, because the region of interest is typically defined as an isotropic neighbourhood around a point, which does not adapt to the geometry of the shape parts. We introduce the bilateral map, a local shape descriptor whose region of interest is defined by two feature points. Compared to the classical descriptor definition using a single point, the bilateral approach exploits the use of a second point to place more constraints on the selection of the spatial context for feature analysis. This leads to a descriptor where the shape of the region of interest adapts to the context of the two points, making it more refined for shape matching. In particular, we show that our new descriptor is more effective for partial matching, because potentially extraneous regions of the models are selectively ignored owing to the adaptive nature of the bilateral map. This property also renders the bilateral map partially insensitive to topological changes. We demonstrate the effectiveness of the bilateral map for partial matching via several correspondence and retrieval experiments and evaluate the results both qualitatively and quantitatively.