Rectifying developable surfaces of framed base curves and framed helices

Flat surfaces along cuspidal edges

“…Remark 2.6. We remark that invariants of frames along curves in R 3 with singularities (framed singular curves) are studied in [10]. The frame considered in the present paper is constructed by using the normal vector of given frontal.…”

Evolutes and focal surfaces of framed immersions in the Euclidean space

“…is constant if and only if γ is a framed helix, see [9]. Therefore, if M (t) = ±L(t) for all t ∈ I, then γ is a framed helix.…”

Singularities of Non-Developable Surfaces in Three-Dimensional Euclidean Space

“…In this section, we study the ruled surface generated by Frenet-type framed base curves (see [15][16][17]). Since there exist singular points on these surfaces, in general, we cannot construct the normal vector of these surfaces.…”

“…If there exists a regular unit speed curve T : Y → S 2 and a C ∞ function ι : Y → R satisfieṡ β(y) = ι(y)T(y) for all y ∈ Y, then we call β = β(y) : Y → R 3 a Frenet -type framed base curve (FTFB curve). We call {T(y), N(y), B(y)} an orthonormal frame along β(y) in R 3 , where N =Ṫ/ Ṫ and B = T × N. More details are available from [16]. The FTFB curve is one special kind of framed base curve [15,16].…”

“…We call {T(y), N(y), B(y)} an orthonormal frame along β(y) in R 3 , where N =Ṫ/ Ṫ and B = T × N. More details are available from [16]. The FTFB curve is one special kind of framed base curve [15,16]. As we want to intuitively observe the properties at singular points on the ruled surface, we choose this kind curve that is similar to the Frenet curve.…”