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

Harmonic Evolute Surface of Tubular Surfaces via B -Darboux Frame in Euclidean 3-Space E. M. Solouma Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, Saudi Arabia Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Egypt https://orcid.org/0000-0001-5000-2061 Ibrahim AL-Dayel Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, Saudi Arabia https://orcid.org/0000-0002-5901-2511 Manuel De Le n

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.
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