To date, algorithms designed to recognize 3-d objects from intensity images have either employed global features or local features based on straight line segments. Global features limit recognition to objects that are completely visible and segmented from the background; these are often unrealistic conditions in machine vision domains. Local features based on line segments are somewhat more flexible since partially visible objects may be recognized, however, the requirement that objects will have sufficient straight lines to ensure robust recognition is also limiting. In this paper, we discuss a set of features, called scale-invariant critical point neighborhoods, or SICPN's, which are more generally applicable to 3-d object recognition. SICPN's efficiently encode the local shape of edge segments near points of high curvature. To within variations caused by image noise and discretization, SICPN's are invariant to image plane translations, rotations, and scaling. These invariant properties enhance the utility of these features in 3-d recognition algorithms. Furthermore, we show that SICPN's have many other desirable characteristics including informativeness, ease of detection, and compact representation. In addition, we empirically demonstrate that SICPN's are insensitive to noise. The above characteristics are essential if a feature is to be useful for 3-d object recognition.