Abstract 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.

中文翻译：

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Lorentz-Minkowski 3 空间中圆和类空间曲线的包络

摘要 各向同性投影建立了洛伦兹-闵可夫斯基空间 𝐄 1 3 {\mathbf{E}_{1}^{3}} 中的曲线与欧几里得平面中的循环族（即拉盖尔平面中的曲线 ℒ 2 {\mathcal{L}^{2}}）。在本文中，我们将根据众所周知的 𝐄 1 3 {\mathbf{E}_{ 平滑曲线的洛伦兹不变量，给出两个给定的循环族通过（扩展的）拉盖尔变换相关的充分必要条件1}^{3}}。我们将讨论 𝐄 1 3 {\mathbf{E}_{1}^{3}} 中单位速度类空曲线的二阶导数的因果特征，根据相应的定向圆族及其包络的几何形状。研究了几个包络线是众所周知的平面曲线的圆族，并计算了它们的拉盖尔不变量。