arXiv subjects

Nozomi Nakatsuyama

Publications and source records attributed to Nozomi Nakatsuyama.

10 recordsLinked to original sources

arXiv2026

Curvature equivalence for Legendre curves in the unit tangent bundle over Euclidean plane

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 paper, we introduce an equivalence relation for Legendre curves called the curvature equivalence. We investigate properties of the curvature equivalence and give local and global classifications of Legendre curves under the curvature equivalence.

math.DG↗

arXiv2026

Evolutes, involutes and parallels of Legendre curves in hyperbolic and de Sitter 2-spaces via rotational transformations

We develop a unified approach to constructing the moving frames of the evolute, involute and parallel of a Legendre curve via rotational transformations (rotation, boost and null rotation). The definitions of the evolute, involute and parallel of a Legendre curve in hyperbolic and de Sitter 2-spaces are systematically presented. Furthermore, the connections and dualities among these curves are discussed.

math.DG↗

arXiv2026

Horocyclic evolutes, parallels and involutes of spacelike frontals in hyperbolic 2-space

The horocyclic evolutes of spacelike frontals in hyperbolic 2-space have already been defined. Using enveloid theorem, we now define the horocyclic parallel and involute of a spacelike frontal in hyperbolic 2-space as the normal envelopes of its normal and tangent horocycles, respectively. Meanwhile, we investigate the relations among horocyclic evolutes, parallels and involutes.

math.DG↗

arXiv2026

Evolutes and involutes of framed curves in Euclidean 3-space

We investigated the evolute of a space curve with singular points. As smooth curves with singular points, we apply the theory of framed curves. However, the involute corresponding to the evolute in the sense of the locus of the centre of osculating spheres has not been defined as far as we know. In this paper, we directly define the evolutes and involutes of non-degenerate curves and framed curves using the theory of Bertrand type curves. We give conditions that the evolutes and involutes are inverse operations of these curves. Moreover, we investigate the other cases in which the evolutes and involutes are not inverse operations.

math.DG↗

arXiv2026

Bertrand Legendre curves in the unit tangent bundle over Euclidean plane

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 and involute curves, as well as evolutoid and involutoid curves. Since associated curves may have singular points even if the original curve is regular, Legendre curves provide a suitable framework for investigating the properties of such curves. We give existence conditions of Bertrand regular plane curves and Bertrand Legendre curves. Moreover, we give an inverse operation for Bertrand Legendre curves. Furthermore, we define a mapping between sets of Legendre curves using Bertrand Legendre curves and prove that this mapping is bijective up to equivalence relations.

math.DG↗

arXiv2025

Singularities of translation surfaces under the linearly dependent condition in Euclidean 3-space

We investigated singular points of translation surfaces under the linearly independent condition. In this paper, as completion, we investigate singular points of translation surfaces under the linearly dependent condition, using the theories of generalised framed surfaces and framed surfaces. We introduce the notion of translation generalised framed surfaces and investigate types of singular points that appear on translation generalised framed base surfaces.

math.DG↗

arXiv2025

Bertrand framed surfaces in the Euclidean 3-space and its applications

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 Bertrand framed surfaces. As applications, we can directly define the caustics and involutes of framed surfaces, and give conditions that the caustics and involutes are inverse operations of framed surfaces like as those of Legendre curves. Moreover, a framed surface is one of the Bertrand framed surfaces if and only if another caustic of the involute exists, under conditions. Furthermore, we find a new such operation, the so-called tangential direction framed surfaces.

math.DG↗

arXiv2024

Bertrand lightcone framed curves in the Lorentz-Minkowski 3-space

We consider mixed types of not only regular curves but also curves with singular points in the Lorentz-Minkowski 3-space. In order to consider mixed type of curves with singular points, we consider the lightcone frame and lightcone framed curves. By using lightcone frame, we can consider Bertrand types for lightcone framed curves, so-called Bertrand lightcone framed curves. We clarify that the existence conditions of Bertrand lightcone framed curves in all cases. As a consequence, we find pseudo-circular involutes and evolutes of mixed type curves which appear as Bertrand lightcone framed curves.

math.DG↗

arXiv2024

On vertices of frontals in the Euclidean plane

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 (respectively, fronts) in the Euclidean plane. We define a vertex using evolutes of frontals. After that we define a vertex of a frontal in the general case. It is also known that the four vertex theorem does not hold for simple closed fronts. We give conditions under which a frontal has a vertex and the four vertex theorem holds for closed frontals. We also give examples and counter examples of the four vertex theorem.

math.DG↗

arXiv2024

Bertrand types of regular curves and Bertrand framed curves in the Euclidean 3-space

A Bertrand (respectively, Mannheim) curve is a space curve whose principal normal line is the same as the principal normal (respectively, bi-normal) line of another curve. By definition, another curve is a parallel curve with respect to the direction of the principal normal vector. In this paper, we consider the other cases, that is, a space curve whose tangent (or, principal normal, bi-normal) line is the same as the tangent (or, principal normal, bi-normal) line of another curve, respectively. We say that a Bertrand type curve if there exists such another curve. We clarify that the existence conditions of Bertrand type curves in all cases. There are times when the Bertrand type curve does not exist. On the other hand, since the another curve may have singular points, we also consider curves with singular points. As smooth curves with singular points, it is useful to use the framed curves in the Euclidean space. Then we define and investigate Bertrand framed curves. We also clarify that the existence conditions of the Bertrand framed curves in all cases.

math.DG↗