Related papers: Legendre curves and singularities of a ruled surfa…
The aim of this paper is to give some characterizations for N-Legendre and N-slant curves in the unit tangent bundles of surfaces endowed with natural diagonal lifted structures.
In this paper, we investigate the ruled surfaces generated by a straight line according to rotation minimizing frame (RMF). Using this frame of a straight line, we obtained the necessary and sufficient conditions when the ruled surface is…
The Legendre curve in the unit tangent bundle over Euclidean plane is a plane curve with a moving frame. We have the (Legendre) curvature of the Legendre curve, and the existence and uniqueness theorems for the curvature are valid. In this…
We investigate not only the associated curves of regular plane curves, but also those of Legendre curves. As associated curves, we consider Bertrand regular plane curves and Bertrand Legendre curves. These curves contain parallel, evolute…
In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the…
It is given the diffeomorphism classification on generic singularities of tangent varieties to curves with arbitrary codimension in a projective space. The generic classifications are performed in terms of certain geometric structures and…
In this paper, we have first given easily the characterization of special curves with the help of the Rotation minimizing frame (RMF). Also, rectifying-type curves are generalized n-dimensional space $R_{n}$.
In this paper, we define a new type of ruled surface called ruled surface by using the alternative frame of a base curve. Then, we study its differential geometric properties such as striction line, distribution parameter, fundamental…
For Legendre curves, we consider surfaces of revolution of frontals. The surface of revolution of a frontal can be considered as a framed base surface. We give the curvatures and basic invariants for surfaces of revolution by using the…
We consider ruled surfaces with finite multiplicity. We study behaviors of the striction curves and the singularities of the ruled surfaces. We also give geometric meanings of invariants related to the ruled surfaces.
Let $M$ be a surface with a Riemannian metric and $UM$ the unit tangent bundle over $M$ with the canonical contact sub-Riemannian structure $D$ on $UM$. In this paper, the complete local classification of singularities, under the Legendre…
We propose a unified method to visualize curvature on planar curves and surfaces of revolution using the tangential angle parameter. For plane curves, placing markers at equal increments of the tangential angle reveals local bending…
We obtain the explicit representation of Legendre surfaces in the unit $5$-sphere with harmonic mean curvature vector field, under the condition that the mean curvature function is constant along a certain special direction.
Legendre curves are smooth plane curves which may have singular points, but still have a well defined smooth normal (and corresponding tangent) vector field. Because of the existence of singular points, the usual curvature concept for…
We investigate vertices for plane curves with singular points. As plane curves with singular points, we consider Legendre curves (respectively, Legendre immersions) in the unit tangent bundle over the Euclidean plane and frontals…
Directional fields, including unit vector, line, and cross fields, are essential tools in the geometry processing toolkit. The topology of directional fields is characterized by their singularities. While singularities play an important…
In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.
This thesis is devoted to the Differential Geometry of curves and surfaces along with applications in Quantum Mechanics. In the 1st part we introduce the well known Frenet frame. Later, we show that the curvature function is a lower bound…
In this paper, we consider Legendre trajectories of trans-$S$-manifolds. We obtain curvature characterizations of these curves and give a classification theorem. We also investigate Legendre curves whose Frenet frame fields are linearly…
In this paper, we study the curvature properties of random complex plane curves. We bound from below the probability that a uniform proportion of the area of a random complex degree $d$ plane curve has a curvature smaller than $-d/8$. Our…