Related papers: Relatively normal-slant helices lying on a surface…
We consider the Anti-de Sitter space $\mathbb{H}^3_1$ equipped with Berger-like metrics, that deform the standard metric of $\mathbb{H}^3_1$ in the direction of the hyperbolic Hopf vector field. Helix surfaces are the ones forming a…
We characterize helix surfaces (constant angle surfaces) in the special linear group $\mathrm{SL}(2,\r)$. In particular, we give an explicit local description of these surfaces in terms of a suitable curve and a 1-parameter family of…
We consider a unit speed curve $\alpha$ in Euclidean $n$-dimensional space $E^n$ and denote the Frenet frame by $\{v_1,...,v_n\}$. We say that $\alpha$ is a cylindrical helix if its tangent vector $v_1$ makes a constant angle with a fixed…
The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion.…
We consider ruled surfaces in the three-dimensional Euclidean space and some geometrically distinguished families of curves on them whose normal curvature has a concrete form. The aim of this paper is to find and classify all ruled surfaces…
This paper deals with relative normalizations of skew ruled surfaces in the Euclidean space $\mathbb{E}^{3}$. In section 2 we investigate some new formulae concerning the Pick invariant, the relative curvature, the relative mean curvature…
A framed surface is a smooth surface in the Euclidean space with a moving frame. By using the moving frame, we can define Bertrand framed surfaces as the same idea as Bertrand framed curves. Then we find the caustics and involutes as…
In this paper, the spinor formulation of Darboux frame on an oriented surface is given. Also, the relation between the spinor formulation of Frenet frame and Darboux frame are obtained.
The main objective of the present paper is to investigate a sufficient condition for which a rectifying curve on a smooth surface remains invariant under isometry of surfaces, and also it is shown that under such an isometry the component…
We investigate helicoidal (screw) surfaces generated not only by regular curves but also by curves with singular points. For curves with singular points, it is useful to use frontals in the Euclidean plane. The helicoidal surface of a…
The present paper deals with some characterizations of rectifying and osculating curves on a smooth surface with respect to the reference frame $\{\vec{T},\ \vec{N},\ \vec{T}\times\vec{N}\}$. We have computed the components of position…
Considering the tangent plane at a point to a surface in the four-dimensional Euclidean space, we find an invariant of a pair of two tangents in this plane. If this invariant is zero, the two tangents are said to be conjugate. When the two…
In this article a relation between curvature functionals for surfaces in the Euclidean space and area functionals in relative differential geometry will be given. Relative differential geometry can be described as the geometry of surfaces…
This paper is devoted to the 3-dimensional relative differential geometry of surfaces. In the Euclidean space $\R{E} ^3 $ we consider a surface $\varPhi %\colon \vect{x} = \vect{x}(u^1,u^2) $ with position vector field $\vect{x}$, which is…
In this paper, we introduce isophote curves on surfaces in Galilean 3-space. Apart from the general concept of isophotes, we split our studies into two cases to get the axis d of isophote curves lying on a surface such that d is an…
The ruled surfaces, i.e., surfaces generated by one parametric set of lines, are widely used in the~field of applied geometry. An~isophote on a surface is a curve consisting of surface points whose normals form a constant angle with some…
In this paper, we introduce a new class of curves \alpha called a f-rectifying curves, which its f-position vector defined by {\alpha}_{f}(s)=\int f(s)T(s)ds always lie in the rectifying plane of \alpha, where f is an integrable function…
The aim of this paper is to investigate the sufficient condition for the invariance of a normal curve on a smooth immersed surface under isometry. We also find the the deviations of the tangential and normal components of the curve with…
We investigate the behaviour of vertices and inflexions on 1-parameter families of curves on smooth surfaces in the 3-space, which include a singular member. In particular, we discuss the context where the curves evolve as sections of a…
In this paper, we take into account the opinion of involute-evolute curves which lie on fully surfaces and by taking into account the Darboux frames of them we illustrate these curves as special involute-evolute partner D-curves in E3.…