Akademik Veri Yönetim Sistemi
NEW CONCEPTS AND ADVANCED STUDIES IN NATURAL SCIENCE AND MATHEMATICS , Prof. Dr. Canan ÖZDEMİR, Editör, ALL SCIENCES ACADEMEY, Konya, ss.220-235, 2025
In this study, we investigate the geometric characteristics of
canal-type surfaces defined in the Euclidean three-dimensional space using the
Darboux frame. Surfaces formed by the motion of a sphere along a spine curve,
commonly known as canal surfaces, possess rich differential geometric structures. The Darboux frame is a
fundamental tool in differential geometry used to analyze the local geometry of
curves lying on the surface. As a surface-adapted version of the Frenet frame,
it takes into account the tangent direction of the curve together with the
surface-tangent and surface-normal components. This framework enables a
detailed study of geodesic, asymptotic, and curvature lines, offering deeper
insight into the intrinsic geometry of surfaces. First, the fundamental
forms (from the first to the third) of the canal surface expressed in the
Darboux frame are obtained, along with the corresponding curvature quantities. And, we focus on Weingarten
surfaces, including linear Weingarten canal surfaces, and analyze their
characteristics in terms of curvature relations. Additionally,
we examine developable and minimal canal surfaces, determining the conditions
that characterize their existence. To validate our theoretical findings, we present a concrete example
constructed with specific parametric equations. This example illustrates how
the derived conditions manifest in practice and confirms the applicability of
our approach. The results contribute to the broader study of surface theory and
may be useful in fields such as computer-aided geometric design and theoretical
physics.
Keywords – Euclidean 3-space, Canal surfaces, Darboux
frame, Weingarten surfaces, Linear Weingarten surfaces