Miura and Miura-derivative rigid origami patterns are increasingly used for engineering and architectural applications. However, geometric modelling approaches used in existing studies are generally haphazard, with pattern identifications and parameterizations varying widely. Consequently, relationships between Miura-derivative patterns are poorly understood, and widespread application of rigid patterns to the design of folded plate structures is hindered. This paper explores the relationship between the Miura pattern, selected because it is a commonly used rigid origami pattern, and first-level derivative patterns, generated by altering a single characteristic of the Miura pattern. Five alterable characteristics are identified in this paper: crease orientation, crease alignment, developability, flat-foldability, and rectilinearity. A consistent parameterization is presented for five derivative patterns created by modifying each characteristic, with physical prototypes constructed for geometry validation. It is also shown how the consistent parameterization allows first-level derivative geometries to be combined into complex piecewise geometries. All parameterizations presented in this paper have been compiled into a matlab Toolbox freely available for research purposes.