Why do some pairs of short line segments look special and others random? Minimal Model Theory (MMT) holds that two-line configurations are more special if there are fewer degrees of freedom left to change their relative position and orientation. Another theory emphasizes the role of nonaccidental properties (NAPs) that allow 3D shape recovery. Predictions from both theories were tested in three experiments with displays consisting of four two-line configurations, three identical and one different, which had to be located. The odd one out differed from the others by a change in angle or position (e.g., from T- to L- or X-junction) in two experiments. In Experiment 3, metric changes were directly compared to changes in NAPs such as collinearity cotermination, and parallelism. Overall, changes in NAPs were detected faster than metric changes, while position and orientation changes in junctions were not always as predicted by MMT.