Envelopes of One-Parameter Family of Frontals Related to Two Classes of Legendrian Dualities

Giving the definitions of one-parameter family of frontals in lightcone and one-parameter family of spacelike Legendrian curves in Δ2 and Δ3, and further using the variability condition and the tangency condition, the definitions of envelopes of these geometric objects are presented. The aim of this work is to explore the criterion conditions on the envelopes of a one-parameter family of frontals related to Δ2 and Δ3 in three pseudo-spheres. Thereby the characterizations of these envelopes are described via Envelope Theorems. The geometric relations among these envelopes are discussed in detail. It is demonstrated that the Δ2-duality or the Δ3-duality of one-parameter family of frontals among three pseudo-spheres leads to the fact that the one-parameter family of frontals that are Δ2-duality or Δ3-duality each other share the same envelope. In addition, the hyperbolic and de Sitter parallels of the one-parameter family of spacelike frontals are also defined, and the existence conditions of the envelopes of such parallels are investigated correspondingly. Finally, two examples are provided to understand the theoretical results.