The isotropy projection establishes a correspondence between curves in the Lorentz–Minkowski space and families of cycles in the Euclidean plane (i.e., curves in the Laguerre plane ). In this paper, we shall
give necessary and sufficient conditions for two given families of cycles to be related by a (extended) Laguerre transformation in terms of the well known Lorentzian invariants for smooth curves in . We shall discuss the causal character of the second derivative of unit speed spacelike curves in in terms of the geometry of the corresponding families of oriented circles and their envelopes. Several families of circles whose envelopes are well-known plane curves are investigated and their Laguerre invariants computed.