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

Abstract

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 ( $\mathbb{B}$ -Darboux frame) in Euclidean 3-space ${\text{E}}^3$ to study and explain the geometric characteristics of the harmonic evolute surfaces of tubular surfaces. The characterizations of the harmonic evolute surface’s $\varrho$ and $\varsigma$ 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.