We establish a theoretical link between 3D edge detection and local surface approximation using uncertainty. As a practical application of the theory, we present a method for computing typical curvature features from 3D medical images. We determine the uncertainties inherent in edge (and surface) detection in 2- and 3-dimensional images by quantitatively analysing the uncertainty in edge position, orientation and magnitude produced by the multidimensional (2D and 3D) versions of the Deriche-Canny recursive separable edgedetector. The uncertainty is shown to depend on edge orientation, e.g. the position uncertainty may vary with a ratio larger than 2.8 in the 2D case, and 3.5 in the 3D case. These uncertainties are then used to compute local geometric models (quadric surface patches) of the surface, which are suitable for reliably estimating local surface characteristics; for example, Gaussian and Mean curvature. We demonstrate the effectiveness of our methods compared to previous techniques. These curvatures are then used to obtain more structured features such as curvature extrema and lines of curvature extrema. The final goal is to extract robust geometric features on which registration and/or tracking procedures can rely.