In this study, we introduce a new type of surface curves called $D$-type curve. This curve is defined by the property that the unit Darboux vector $vec{W}_{0} $ of a surface curve $vec{r}(s)$ and unit surface normal $vec{n} $ along the curve $vec{r}(s)$ satisfy the condition $leftlangle vec{n} ,vec{W}_{0} rightrangle =text{constant}$. We point out that a $D$-type curve is a geodesic curve or an...

This paper studies the problem of designing a rational Bézier developable surface pencil with a common isogeodesic, and provides an algorithm for the representation of complicated geometric models in industrial applications which need to satisfy that the shape surface can be developed and a given curve is geodesic. By employing the local Frenet orthonormal frame, the explicit expression of the ...

In this paper, we study the bicanonical pencil of a Godeaux surface and of a determinantal Barlow surface. This study gives a simple proof for the unobstructedness of deformations of a determinantal Barlow surface. Then we compute the number of hyperelliptic curves in the bicanonical pencil of a determinantal Barlow surface via classical Prym theory.

In the present note we show that any surface of general type over C with pg = 2,q = 1 and non birational bicanonical map has a pencil of curves of genus 2. Combining this result with previous ones, one obtains that an irregular surface S of general type with χ(S) ≥ 2 and non-birational bicanonical map has a pencil of curves of genus 2.

This paper proposes an approach to design the surface pencil with specified parametric representations from a given geodesic. The surface was considered as a combination of the marching-scale functions and the Frenet trihedron frame, for which some necessary and sufficient conditions must be met. This paper simplifies those conditions, and represents the surface by combination of control points...

In this paper, we study the problem of constructing a family of surfaces from a given spatial geodesic curve. We derive a parametric representation for a surface pencil whose members share the same geodesic curve as an isoparametric curve. By utilizing the Frenet trihedron frame along the given geodesic, we express the surface pencil as a linear combination of the components of this local coord...

Two-dimensional (2D) materials are widely used in various applications due to their extraordinary properties. In particular, electrochemical stability, low electrical resistance, and huge specific surface area make them very interesting active for supercapacitors. Herein, flexible, biodegradable, low-cost supercapacitors introduced a simple way based on hand-drawing pencil traces or -rubbing mo...