Envelopes of Legendre Curves in the Unit Tangent Bundle over the Euclidean Plane

Abstract

For singular plane curves, the classical definitions of envelopes are vague. In order to define envelopes for singular plane curves, we introduce a one-parameter family of Legendre curves in the unit tangent bundle over the Euclidean plane and the curvature. Then we give a definition of an envelope for the one-parameter family of Legendre curves. We investigate properties of the envelopes. For instance, the envelope is also a Legendre curve. Moreover, we consider bi-Legendre curves and give a relationship between envelopes.

Envelopes of legendre curves in the unit spherical bundle over the unit sphere

“…We first recall the definition of the envelope of a one-parameter family of Legendre curves in the unit tangent bundle over the Euclidean plane. For more detailed descriptions see [11].…”

“…Since e : U → I × Λ is a pre-envelope of ( γ, ν), we have γ λ (e(u)) • ν(e(u)) = 0 for all u ∈ U (cf. [11]). It follows that…”

“…Therefore e : U → I × Λ is a pre-envelope of ( γ, ν) (cf. [11]). Moreover, we have E γ (u) = γ • e(u) = π • γ • e(u) = π(E γ (u)) = E γ (u) for all u ∈ U .…”

“…On the other hand, for singular plane curves, the classical definitions of envelopes are vague. In [11], the third author clarified the definition of the envelope for a family of singular curves in the unit tangent bundle over the Euclidean plane. This idea can be generalized to an envelope of a family of singular spherical curves.…”

Dualities and envelopes of one‐parameter families of frontals in hyperbolic and de Sitter 2‐spaces

“…We have to develop new methods to study envelopes of singular curves. The last author had introduced an effective way to investigate an envelope of a one‐parameter family of a special kind of singular curves, namely, frontals in the Euclidean plane [5, 12]. We can also define the frontals in hyperbolic and de Sitter 2‐spaces [3], respectively, by using the Legendrian dualities developed in [2, 7, 8].…”