LAUSR

Abstract The isotropy projection establishes a correspondence between curves in the Lorentz–Minkowski space ????13{\mathbf{E}_{1}^{3}} and families of cycles in the Euclidean plane (i.e., curves in the Laguerre plane ℒ2{\mathcal{L}^{2}}). 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 ????13{\mathbf{E}_{1}^{3}}. We shall discuss the causal character of the second derivative of unit speed spacelike curves in ????13{\mathbf{E}_{1}^{3}} 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.