Catastrophe-theory-based Autonomous Terrain-Feature UAV Relative (CATFUR) navigation is geolocation without the Global Positioning System (GPS). As fully autonomous navigation based only on recognition of terrain features, it can be integrated with GPS or other state-of-the-art navigation systems, or can be independent. CATFUR navigation is based on integration/comparison and sensor fusion of DEM (digital elevation map) 3-D data, processed by commercial off-the-shelf geographic information system (COTS GIS) environments into a vectorial graph. CATFUR obtains data from the vertical takeoff unmanned air vehicle (VTUAV) COTS inertial and visual sensors, and from components of an azimuth-elevation local positioning system (LPS). Real-time data processing could perform on highly parallel 2 in. x 3 in. Application specific hardware. Typical point or line catastrophic singularities on surfaces are edges, ridges, wrinkles, and surface cracks. Such singularities have a fixed location on the surface. In contrast, catastrophes have the unexpected property of not being fixed to a surface. Catastrophes can be the basis of GPS-independent relative navigation, based only on the existence of a folded terrain, even without landmarks. Since mountains do not move, we can use mountain guidance, much as star guidance has been used for centuries to navigate the oceans.