Evolutes and focal surfaces of framed immersions in the Euclidean space

Abstract

We consider a smooth curve with singular points in the Euclidean space. As a smooth curve with singular points, we have introduced a framed curve or a framed immersion. A framed immersion is a smooth curve with a moving frame and the pair is an immersion. We define an evolute and a focal surface of a framed immersion in the Euclidean space. The evolutes and focal surfaces of framed immersions are generalizations of each object of regular space curves. We give relationships between singularities of the evolutes and of the focal surfaces. Moreover, we consider properties of the evolutes, focal surfaces and repeated evolutes.

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

“…Moreover, parallel curves may have singular points. The locus of the singular points of parallel curves is the evolute of the original curve, see [6,9,10,13]. We consider smooth curves with singular points.…”

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

“…The parallel curve of a framed curve is defined in [9]. We define a parallel curve of a one-parameter family of framed curves.…”

On the trajectory ruled surfaces of framed base curves in the Euclidean space

“…If a given curve has singular points, the Frenet frame of these curves cannot be constructed. In order to construct the Frenet frame for nonregular curves, we need the notion of framed curve and framed base curve 7 . Tangent vectors of a nonregular curve vanish at singular points, so we will consider a regular spherical curve to construct the Frenet frame.…”

“…Then, these curves called Frenet‐type framed base curve. For more details on framed curves and framed surfaces, see previous studies 7–11 …”

“…In order to construct the Frenet frame for nonregular curves, we need the notion of framed curve and framed base curve. 7 Tangent vectors of a nonregular curve vanish at singular points, so we will consider a regular spherical curve to construct the Frenet frame. Then, these curves called Frenet-type framed base curve.…”