arXiv research

The study classifies singularities of spherical orthotomic curves.

Spherical orthotomic curves are equivalent to pedal and dual curves under certain conditions.

The paper characterizes pedal curves of quadratic curves.

The paper studies singularities of pedal curves of hyperbolic frontals.

New Finsler metrics derived from pedal curves.

Pedal curves derived from ellipses are invariant in area.

Study circle families' envelopes and related curves.

A Steiner deltoid maintains constant area across all boundary points of an ellipse.

Study the geometry of a surface formed by extending a Whitney umbrella.

Defines non-parabolic curves in spatial hybrid space with applications.

The calculus correspondence has been known to exist between generic pedal evolutions and generic wave front evolutions. In this paper, we first extend the known results on the calculus correspondence to evolutions with multi-parameters, and then give applications of calculus correspondence. Moreover, we discuss the pos…

In this paper, we introduce the notions of map-germs of pedal unfolding type and normalized Legendrian map-germs; and then we show that the fundamental theorem of calculus provides a natural one to one correspondence between Whitney umbrellas of pedal unfolding type and normalized swallowtails.

Study spherical curves with curvature dependent on distance to a great circle.

The reductivity of a spherical curve represents how reduced the spherical curve is. It is unknown if there exists a spherical curve whose reductivity is four. In this paper we give an unavoidable set for spherical curves with reductivity four by considering 4-gons.

The reductivity of a spherical curve is the minimal number of a local transformation called an inverse-half-twisted splice required to obtain a reducible spherical curve from the spherical curve. It is unknown if there exists a spherical curve whose reductivity is four. In this paper, an unavoidable set of configuratio…

We show that we can obtain a reducible spherical curve from any non-trivial spherical curve by four or less inverse-half-twisted splices, i.e., the reductivity, which represents how reduced a spherical curve is, is four or less. We also discuss unavoidable sets of tangles for spherical curves.

Paper develops invariants for spherical curves using chord diagrams.

In this paper, we give definitions and characterizations of normal and spherical curves in the dual space. We show that normal curves are also spherical curves in D^3.

In this work, we studied the properties of the spherical indicatrices of involute curve of a space curve and presented some characteristic properties in the cases that involute curve and evolute curve are slant helices and helices, spherical indicatrices are slant helices and helices and we introduced new representatio…

The paper explores geometric properties of interception curves on planes and spheres.

New spherical curve deformations solve a conjecture.

Existence and uniqueness of spherical helicoidal surfaces in 3-sphere via spherical curves.

In this work, we studied the properties of the spherical indicatrices of a Bertrand curve and its mate curve and presented some characteristic properties in the cases that Bertrand curve and its mate curve are slant helices, spherical indicatrices are slant helices and we also researched that whether the spherical indi…

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 angle between the principal normal and an RM vector field for spherical curves. L…

Defines new curves from tangent indicatrix of curves, linking them to helices and slant helices.

Study extends geodesic curvature formula to higher dimensions.

The paper studies surfaces with spherical curvature lines and their generation by constrained elastic curves.

We give a complete description of all order 1 invariants of spherical curves. We also identify the subspaces of all J-invariants and S-invariants, and present two equalities satisfied by any spherical curve.

Study null surfaces of pseudo-spherical curves in anti-de Sitter space.

The study proves spherical polygon analogs of curve theorems, finding bounds on intersections and inflections.

Study counts sub-chord diagrams to classify spherical curves.

In this paper, new representations of a Bertrand curve pair in three dimensional Lie groups with bi-invariant metric are given. Besides, the spherical indicatrices of a Bertrand curve pair are obtain and the relations between the spherical indicatrices and new representations of Bertrand curve pair are shown.

Given a simple closed plane curve $Γ$ of length $L$ enclosing a compact convex set $K$ of area $F$, Hurwitz found an upper bound for the isoperimetric deficit, namely $L^2-4πF\leq π|F_{e}|$, where $F_{e}$ is the algebraic area enclosed by the evolute of $Γ$. In this note we improve this inequality finding strictly posi…

Paper extends Schur's theorem to spherical curves via monotonicity.

The study proves that geodesic spherical curves characterize manifolds of constant curvature.

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. In this work, we extend these investigations to characterize curves that lie on a…

In this article, we investigate Bertrand curves corresponding to the spherical images of the tangent, binormal, principal normal and Darboux indicatrices of a space curve in Euclidean 3-space. As a result, in case of a space curve is a general helix, we show that the curves corresponding to the spherical images of its …

Spherical quadrilaterals classified based on geometric properties.

Study spherical curve flow with density in a specific geometric space.

The aim of this paper is to determine criteria of being integral curve for the geodesic spray of the natural lift curves of the spherical indicatrices of the involutes of a given spacelike curve with a timelike binormal in Minkowski 3-space. Furthermore, some interesting results about the spacelike evolute curve with t…

The paper characterizes curves in pseudo-Galilean 4-space.

In this paper, we investigate a curve whose spherical image the tangent indicatrix and binormal indicatrix is slant helix and called it as a slant helix. We obtain that the spherical images are spherical slant helices defined by [3]. This notation is a generalization of a slant helix. Furthermore, we have given some ch…

Study geodesics on spherical polyhedra, estimating their number.

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 regular curves cannot be straightforwardly extended to these curves. However, Fukunag…

We show non-collapsing for the evolution of nearly spherical closed convex curves in \mathbb{R}^2 under power curvature flow using two-point-methods.

Nearly spherical, positively curved surfaces are mapped from a sphere.

Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.

We show that the maximal number of singular moves required to pass between any two regularly homotopic planar or spherical curves with at most n crossings, grows quadratically with respect to n. Furthermore, this can be done with all curves along the way having at most n+2 crossings.