This paper is about the automated discovery and mapping of surfaces using a stereo pair. We begin with the observation that for any workspace which is topologically connected (i.e. does not contain free flying islands) there exists a single surface that covers the entirety of the workspace. We call this surface the covering surface. We assume that while this surface is complex and self intersecting every point on it can be imaged from a suitable camera pose and furthermore that it is locally smooth at some finite scale – it is a manifold. We show how by representing the covering surface as a non- planar graph of observed pixels we are able to plan new views and importantly fuse disparity maps from multiple views. The resulting graph can be lifted to 3D to yield a full scene reconstruction.