Construction and analysis of the quasi-ruled surfaces based on the quasi-focal curves in ${\mathbb{R}}^3$

This paper presents the concept of a quasi-ruled surface, which is a ruled surface generated by a base curve and a ruling, both of which are defined by the quasi-frame (q-frame). This study begins with the original curve defined by the q-frame, and then we focus on the focal curve of the original curve, which serves as the base curve of the ruled surface. We define the focal curve by the q-frame, so the terminology quasi-focal curve is used in this paper. This paper investigates the formation and properties of the quasi-ruled surface (QRS) using a quasi-focal curve (QFC) as the base curve (directrix). The ruling of the surface is expressed in terms of the q-frame associated with the QFC. A variety of QRS types are discussed in this study, including the osculating, normal, and rectifying types. In addition, the types of a quasi-tangent developable surface, a quasi-principal normal surface, and a quasi-binormal ruled surface will also be discussed. The geometric properties of these surfaces, such as the first and second fundamental quantities, Gaussian curvature, mean curvature, second Gaussian curvature, and second mean curvature, are described. The conditions for their developability and minimality are derived. Moreover, we provide an example that includes the study of geometric properties and clear visualizations of these novel types of QRS.