arXiv subjects

Masatomo Takahashi

Publications and source records attributed to Masatomo Takahashi.

17 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

Envelopes created by sphere families in Euclidean 3-space

In this paper, on envelopes created by sphere families in Euclidean 3-space, all four basic problems (existence problem, representation problem, problem on the number of envelopes, problem on relationships of definitions) are solved.

math.DG↗

arXiv2026

Envelopes created by curve families in the Lorentz-Minkowski plane

Classical definitions of envelopes are vague for singular curves in the Lorentz-Minkowski plane. To study envelopes of such curves, we introduce a family of smooth curves that may admit singular points. Notably, the curves considered here may contain both non-lightlike and lightlike points. We then propose a new definition of envelopes for such curves and investigate their properties using lightlike tangential data. Furthermore, we examine the contact between the curves and their envelopes at singular points.

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

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↗

arXiv2026

Hyperbolic generalized framed surfaces in hyperbolic 3-space

Generalizing both hyperbolic framed surfaces and one-parameter families of hyperbolic framed curves, we introduce the concept of hyperbolic generalized framed surfaces and establish their relations in hyperbolic 3-space. We provide the necessary and sufficient conditions for a smooth surface to be a hyperbolic generalized framed base surface, followed by an analysis of the singularities of hyperbolic generalized framed base surfaces. Additionally, relations between hyperbolic generalized framed surfaces, generalized framed surfaces and lightcone framed surfaces are explored. As an application of hyperbolic generalized framed surfaces, we investigate the properties of horocyclic surfaces.

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

Inverse curvature flow of closed Legendre curves

In this paper, we deal with an inverse curvature flow of $\ell$-convex Legendre curves. Since the Legendre curve is a natural generalization of regular curve, the flow is a generalization of the classical inverse curvature flow of regular curves. For the initial value problem, we study on the unique existence of the flow in global time, the monotonicity of the number of the singular cusps with respect to t > 0 and the asymptotic behavior of the flow as $t \to \infty$. Regarding the asymptotics, the flow asymptotically converges to one of the self similar solutions by scaling appropriately, and the convergence is completely categorized depending on the initial curve.

math.AP↗

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↗

arXiv2022

Bertrand and Mannheim curves of framed curves in the 4-dimensional Euclidean space

A Bertrand curve in the 4-dimensional Euclidean space is a space curve whose first normal line is the same as the first normal line of another curve. On the other hand, a Mannheim curve in the 4-dimensional Euclidean space is a space curve whose first normal line is the same as the second or third normal line of another curve. By definitions, another curve is a parallel curve with respect to the direction of the first normal vector. As smooth curves with singular points, we consider framed curves in the Euclidean space. Then we define and investigate Bertrand and Mannheim curves of framed curves. We give necessary and sufficient conditions of Bertrand and Mannheim curves of both regular and framed curves. It is well-known that the Bertrand curves of regular curves do not exist under a condition. However, even if regular curves, Bertrand curves exist as framed curves.

math.DG↗

arXiv2021

Circular evolutes and involutes of framed curves in the Euclidean space

We introduce circular evolutes and involutes of framed curves in the Euclidean space. Circular evolutes of framed curves stem from the curvature circles of Bishop directions and singular value sets of normal surfaces of Bishop directions. On the other hand, involutes of framed curves are direct generalizations of involutes of regular space curves and frontals in the Euclidean plane. We investigate properties of normal surfaces, circular evolutes, and involutes of framed curves. We can observe that taking circular evolutes and involutes of framed curves are opposite operations under suitable assumptions, similarly to evolutes and involutes of fronts in the Euclidean plane. Furthermore, we investigate the relations among singularities of normal surfaces, circular evolutes, and involutes of framed curves.

math.DG↗

arXiv2019

Geometric equivalence among smooth map germs

We consider equivalence relations among smooth map germs with respect to geometry of G-structures on the target space germ. These equivalence relations are natural generalization of right-left equivalence (i.e., A-equivalence) in the sense of Thom-Mather depending on geometric structures on the target space germ. Unfortunately, these equivalence relations are not necessarily geometric subgroups in the sense of Damon (1984). However, we have interesting applications of these equivalence relations.

math.DG↗

arXiv2018

Surfaces of revolution of frontals in the Euclidean space

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 curvatures of Legendre curves. Moreover, we give properties of surfaces of revolution with singularities and cones.

math.DG↗