In this study, we introduce partner ruled surfaces according to the Flc frame that is defined on a polynomial curve. First, the conditions of each couple of two partner ruled surfaces to be simultaneously developable and minimal are investigated. Then, the asymptotic, geodesic and curvature lines of the parameter curves of the partner ruled surfaces are simultaneously characterized. Finally, the examples of the partner ruled surfaces are given, and their graphs are drawn.

In this study, we explore the sweeping surfaces in Euclidean 3-space, utilizing the modified orthogonal frames with non-zero curvature and torsion, which allows us to consider the spine curves even if their second differentiations vanish. If the curvature of the spine curve of a sweeping surface has discrete zero points, the Frenet frame might undergo a discontinuous change in orientation. Therefore, the conventional parametrization with the Frenet frame of such a surface cannot be given. Thus, we introduce two types of modified sweeping surfaces by considering two types of spine curves; the first one’s curvature is not identically zero and the second one’s torsion is not identically zero. Then, we determine the criteria for classifying the coordinate curves of these two types of modified sweeping surfaces as geodesic, asymptotic, or curvature lines. Additionally, we delve into determining criteria for the modified sweeping surfaces to be minimal, developable, or Weingarten. Through our analysis, we aim to clarify the characteristics defining these surfaces. We present graphical representations of sample modified sweeping surfaces to enhance understanding and provide concrete examples that showcase their properties.

In this study, we investigate Smarandache curves constructed by a space curve with a modified orthogonal frame. Firstly, the relations between the Frenet frame and the modified orthogonal frame are summarized. Later, the Smarandache curves based on the modified orthogonal frame are obtained. Finally, the tangent, normal, binormal vectors and the curvatures of the Smarandache curves are determined. A special curve known as the Gerono lemniscate curve whose curvature is not differentiable, the principal normal and binormal vectors are discontinuous at zero point is considered as an example, and the Smarandache curves of this curve are obtained by the aid of its modified orthogonal frame, and their graphics are given.

In this study, we investigate the surfaces created by the movement of the profile curves through the regular polynomial spine curves. To overcome the restrictions of establishing a frame of the polynomial curves at the points where the second and higher-order derivatives vanish, the Frenet-like curve (Flc) frame is considered. In this way, by introducing sweeping surfaces defined based on the Flc frame, we analyze their parameter curves to determine conditions to be geodesics, asymptotics, and principal curvature lines. Furthermore, we derive conditions of these sweeping surfaces to be minimal, developable, and Weingarten surfaces. Lastly, we provide some examples of these sweeping surfaces and illustrate their graphical representations.

In this paper, we present the evolutions of ruled surfaces generated by the principal normal, the principal normal’s derivative, and the Darboux vector fields along a space curve that are the elements of an alternative frame. The comprehension of an object’s rotational behavior is crucial knowledge relevant to various realms, and this can be accomplished by analyzing the Darboux vector along the path of a point on the object as it moves through space. In that regard, examining the evolutions of the ruled surfaces based on the changes in their directrices, including the Darboux vector in the alternative frame along a space curve, is significant. As the first step of this study, we express the evolution of the alternative frame elements of a space curve. Subsequently, the conditions for the ruled surfaces generated by them to be minimal, developable, and inextensible are investigated. These findings can allow some physical phenomena to be well understood through surface evolutions satisfying these conditions. In the final step, we provide graphical representations of some examples of inextensible ruled surfaces and curve evolutions.