We present an efficient grouping strategy for the detection of regular repetitions of planar (but not necessarily coplanar) patterns. At the heart of our system lie the fixed structures that typify the geometric transformations of the regularities. The approach unifies a number of grouping types that have traditionally been dealt with separately. It avoids the use of combinatorics in the search for pattern repetitions, through the combined use of invariants for hashing-based pattern matching on the one hand, and Hough transforms for the detection of the fixed structures on the other hand. In this paper we concentrate on planar homologies and elations in particular. Results on real-world scenes demonstrate the performance of the approach.