<title>Groups for grouping</title>

Abstract

Comparing shapes is important for both recognition and geometry-based grouping. In the case of perspective views of planar shapes, such comparison will typically be based on projective invariants. Yet there are quite a number of practical cases where one can do better and exploit simpler invariants of subgroups of the planar projectivities. This paper tries to sketch a systematic approach, based on subgroups defined by the structures they keep fixed in the image. In particular, this paper focuses on fixed points, fixed lines, lines of fixed points, and pencils of fixed lines. A complete classification of the resulting subgroups is given and the most interesting cases are identified. For these cases invariants are given and examples illustrate their use. In order to illustrate the wider scope, perspectively skewed mirror symmetry is discussed, since it entails a different type of fixed structures.