Envelopes of one-parameter families of framed curves in the Euclidean space

Abstract

As a one-parameter family of singular space curves, we consider a one-parameter family of framed curves in the Euclidean space. Then we define an envelope for a one-parameter family of framed curves and investigate properties of envelopes. Especially, we concentrate on oneparameter families of framed curves in the Euclidean 3-space. As applications, we give relations among envelopes of one-parameter families of framed space curves, one-parameter families of Legendre curves and oneparameter families of spherical Legendre curves, respectively.

One‐parameter families of Legendre curves and plane line congruences

“…is also a one-parameter family of framed curves with respect to 𝑢, where 𝒙(𝑢, 𝑣) = (𝑓(𝑢, 𝑣), 𝑣), ν1 (𝑢, 𝑣) = (𝜈(𝑢, 𝑣), 0) and ν2 (𝑢, 𝑣) = (0, 0, 1) (cf. [12,24]). Moreover, (𝒙, 𝒏, 𝒔) ∶ 𝑈 → ℝ 3 × Δ is a framed surface, where [11,12]).…”

“…We gave the existence and uniqueness theorems for one-parameter families of Legendre curves in [24,29]. However, we give here an explicit construction of one-parameter families of Legendre curves by using the curvatures.…”

“…Then $$\begin{equation*} \sqrt {\big (\mbox{\boldmath $x$},\widetilde{\nu }_1,\widetilde{\nu _2}\,\big ):U \rightarrow \mathbb {R}^3 \times \Delta } \end{equation*}$$is also a one‐parameter family of framed curves with respect to u , where $$\mbox{\boldmath $x$}(u,v)=(f(u,v),v), \widetilde{\nu }_1(u,v)=(\nu (u,v),0)$$ and $$\widetilde{\nu }_2(u,v)=(0,0,1)$$ (cf. [12, 24]). Moreover, $$(\mbox{\boldmath $x$},\mbox{\boldmath $n$},\mbox{\boldmath $s$}):U \rightarrow \mathbb {R}^3 \times \Delta$$ is a framed surface, where …”

Surfaces of congruent sections on cylinders

“…В качестве подвижных образующих жестких плоских кривых в основном используют кривые второго порядка [6][7][8]25], но в ряде случаев необходимость требует применения более сложных кривых [22,23,26]. Выбору конгруэнтных кривых могут помочь параллельные исследования геометров [27].…”

“…Кривошапко [7] и др. В указанных работах [3][4][5][6][7][8] авторы уже использовали понятие поверхностей конгруэнтных сечений.…”