Envelopes of circles and spacelike curves in the Lorentz–Minkowski 3-space

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