Framed curves in the Euclidean space

Abstract

A framed curve in the Euclidean space is a curve with a moving frame. It is a generalization not only of regular curves with linear independent condition, but also of Legendre curves in the unit tangent bundle. We de ne smooth functions for a framed curve, called the curvature of the framed curve, similarly to the curvature of a regular curve and of a Legendre curve. Framed curves may have singularities. The curvature of the framed curve is quite useful to analyse the framed curves and their singularities. In fact, we give the existence and the uniqueness for the framed curves by using their curvature. As applications, we consider a contact between framed curves, and give a relationship between projections of framed space curves and Legendre curves.

Existence conditions of framed curves for smooth curves

“…Note that this curve is not a frontal (see [4,6]). However, we can construct a framed curve , 1)) by using the method in the proof of Theorem 1.2, since the singular set Σ(γ) = {0} and the limit of the derivatives of the tangent vectors exists at the origin.…”

“…We have shown that analytic curves are at least locally framed base curves in the cases of plane curves (n = 2) and space curves (n = 3), see [4] and [6], respectively.…”

“…We denote the set of singular points of γ by Σ(γ), that is, we set Σ(γ) = {t ∈ I |γ(t) = 0}. The framed curves can be characterised by the moving frame {ν(t), µ(t)} of the framed base curve γ(t) and the curvature of the framed curve, in detail see [6].…”

Generic Properties of Framed Rectifying Curves

“…Let R 3 be the three-dimensional Euclidean space, and let γ : I → R 3 be a curve with singular points. In order to investigate this curve, we will introduce the framed curve (cf., [14]). We denote the set ∆ 2 as follows:…”

“…If space curves have singular points, the Frenet frames of these curves cannot be constructed. However, S. Honda and M. Takahashi [14] gave the definition of framed curves. Framed curves are space curves with moving frames, and they may have singular points.…”