Harmonic Evolute Surface of Tubular Surfaces via B -Darboux Frame in Euclidean 3-Space

In this article, we look at a surface associated with real-valued functions. The surface is known as a harmonic surface, and its unit normal vector and mean curvature have been used to characterize it. We use the Bishop-Darboux frame ( B -Darboux frame) in Euclidean 3-space E 3 to study and explain the geometric characteristics of the harmonic evolute surfaces of tubular surfaces. The characterizations of the harmonic evolute surface’s ϱ and ς parameter curves are evaluated, and then, they are compared. Finally, an example of a tubular surface’s harmonic evolute surface is presented, along with visuals of these surfaces.

ON THE HARMONIC EVOLUTE SURFACES OF TUBULAR SURFACES IN EUCLIDEAN 3-SPACE

The aim of this study is to investigate and interpret the geometric properties of the harmonic evolute surfaces of the tubular surfaces in Euclidean 3-space. For this purpose, the harmonic evolute surface is defined by considering the definitions and theorems for the tubular surface constructed in the Euclidean 3-space. The characterizations of the s and ψ parameter curves of the harmonic evolute surface obtained are examined, and then parameter curves of the tubular surface and harmonic evolute surface are compared. Finally, the harmonic evolute surface of a tubular surface is given an example and the graphics of these surfaces are drawn.

Entire solutions of a variation of the eikonal equation and related PDEs

The aim of this paper is twofold. The first aim is to describe the entire solutions of the partial differential equation (PDE) $u_{z_1}^2+2Bu_{z_1}u_{z_2}+u_{z_2}^2=e^g$, where B is a constant and g is a polynomial or an entire function in $\mathbb {C}^2$. The second aim is to consider the entire solutions of another PDE, which is a generalization of the well-known PDE of tubular surfaces.

TUBULAR SURFACES WITH DARBOUX FRAME IN GALILEAN 3-SPACE

In this paper, we define tubular surface by using a Darboux frame instead of a Frenet frame. Subsequently, we compute the Gaussian curvature and the mean curvature of the tubular surface with a Darboux frame. Moreover, we obtain some characterizations for special curves on this tubular surface in a Galilean 3-space.