Groups, fixed sets, symmetries, and invariants

The paper describes a systematic approach to the analysis of symmetries in planar shapes. The shapes can be observed from arbitrary viewpoints. Hence, the results encompass the case of skewed symmetries. In fact, skewing is assumed to arise from perspective distortions, whereas most of the literature restricts the analysis to affine skewing. The point of departure is the identification of structures that symmetries keep fixed in an image. These define subgroups of the projectivities, which in turn assume simpler invariants than their projective counterparts. These invariants allow both simpler and more selective detection and checking of symmetries.