Geometry of generated quasi-ruled surfaces from their quasi-parallel curves according to the q-frame in R3

Parallel curves have been used extensively in a variety of fields during the last decade, including architecture, computer graphics, aerospace, and medicine. Despite these applications, the theoretical development of parallel curves in differential geometry has been relatively limited. In this study, we investigate parallel curves using the q-frame in three-dimensional Euclidean space, rather than the traditional Frenet frame. We introduce the new expression “quasi-parallel curves” as a generalization of classical parallel curves. We also examine several geometric properties of these curves and provide rigorous proofs. In differential geometry, ruled surfaces are of great importance. They are defined by the movement of generators, which generate straight lines on the surface. Furthermore, a directrix (base curve) is any curve that intersects all generators (rulings). In this paper, we investigate a new class of quasi-ruled surfaces whose base curve is both the original curve and its quasi-parallel curve in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . We describe the geometric characteristics of these surfaces and derive the first and second fundamental forms, as well as the Gaussian and mean curvatures. Specific types of quasi-ruled surfaces are analyzed, and conditions for their developability and minimality are established. Several quasi-ruled surfaces generated from the helix curve as a base curve and others generated from its quasi-parallel curve as a base curve are provided, and the parametrization of the resulting surfaces are determined. Finally, all generated surfaces, together with their quasi-parallel and original curves, are visualized using Mathematica 13.