This paper considers the problem of fast and safe autonomous navigation in partially known environments. Our main contribution is a control policy design based on ellipsoidal trajectory bounds obtained from a quadratic state-dependent distance metric. The ellipsoidal bounds are used to embed directional preference in the control design, leading to system behavior that is adapted to the local environment geometry, carefully considering medial obstacles while paying less attention to lateral ones. We use a virtual reference governor system to adaptively follow a desired navigation path, slowing down when system safety may be violated and speeding up otherwise. The resulting controller is able to navigate complex environments faster than common Euclidean-norm and Lyapunov-function-based designs, while retaining stability and collision avoidance guarantees.