We describe a general technique to construct data structures for similarity search in semialgebraic pattern spaces. These spaces capture most known combinations of geometric patterns (e.g., point sets, polygons, polygonal curves) and geometric distance measures for them (e.g. Hausdorff-distance, area of overlap, Fréchet-distance) together with their quotients under various transformation classes (e.g., translations, rigid motions) and they provide the first non-trivial exact search structures in these settings