Generalized normal ruled surface of a curve in the Euclidean 3-space

In this study, we define the generalized normal ruled surface of a curve in the Euclidean 3-space E3. We study the geometry of such surfaces by calculating the Gaussian and mean curvatures to determine when the surface is flat or minimal (equivalently, helicoid). We examine the conditions for the curves lying on this surface to be asymptotic curves, geodesics or lines of curvature. Finally, we obtain the Frenet vectors of generalized normal ruled surface and get some relations with helices and slant ruled surfaces and we give some examples for the obtained results.

Isophote curves on spacelike surfaces in Lorentz–Minkowski space

An isophote curve consists of a locus of surface points whose normal vectors make a constant angle with a fixed vector (the axis). In this paper, we define an isophote curve on a spacelike surface in Lorentz–Minkowski space [Formula: see text] and then find its axis as timelike and spacelike vectors via the Darboux frame. Besides, we give some relations between isophote curves and special curves on surfaces such as geodesic curves, asymptotic curves or lines of curvature.

Normal form of the swallowtail and its applications

We construct a form of the swallowtail singularity in [Formula: see text] which uses coordinate transformation on the source and isometry on the target. As an application, we classify configurations of asymptotic curves and characteristic curves near swallowtails.