Existence conditions of framed curves for smooth curves

Abstract

A framed curve is a smooth curve in the Euclidean space with a moving frame. We call the smooth curve in the Euclidean space the framed base curve. In this paper, we give an existence condition of framed curves. Actually, we construct a framed curve such that the image of the framed base curve coincides with the image of a given smooth curve under a condition. As a consequence, polygons in the Euclidean plane can be realised as not only a smooth curve but also a framed base curve.

Bertrand and Mannheim curves of framed curves in the3-dimensional Euclidean space

“…where µ(t) . We call the mapping We gave the existence and uniqueness theorems for framed curves in terms of the curvatures in [12], also see [11]. Let (γ, ν 1 , ν 2 ) : I → R 3 × ∆ be a framed curve with the curvature (ℓ, m, n, α).…”

Circular evolutes and involutes of framed curves in the Euclidean space

“…On the other hand, we have İγ [t 0 ](t 1 ) = Ïγ [t 0 ](t 1 ) = 0 and rank(I γ [t 0 ] (3) (t 1 ), I γ [t 0 ] (4) (t 1 )) = 2 if and only if α I (t 1 ) = αI (t 1 ) = 0 and αI (t 1 ) = 0. Since equations ( 6), (7) We now consider relations among normal surfaces, circular evolutes, and involutes of framed curves. Let (γ, ν 1 , ν 2 ) : I → R 3 × ∆ be a framed curve with a Bishop frame {v, w, µ}, namely, ℓ(t) = 0 for all t ∈ I in the Frenet-Serret type formula in (1).…”

Evolutes and focal surfaces of framed immersions in the Euclidean space

“…If γ : (I, t 0 ) → R 3 is a real analytic curve germ, then γ is a framed base curve, that is, there exists a mapping germ (ν 1 , ν 2 ) : (I, t 0 ) → ∆ such that (γ, ν 1 , ν 2 ) is a framed curve, see [10]. See also [7].…”