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Atsufumi Honda

Publications and source records attributed to Atsufumi Honda.

At least 19 recordsLinked to original sources
arXiv2025

Isometric deformations of pillow boxes

Pillow boxes are surfaces used for gift boxes, packaging, and even architectural applications. By definition, a pillow box is isometric to a double rectangle consisting of two copies of a rectangle. If the crease pattern is allowed to change, there exist continuous isometric deformations from a pillow box to a double rectangle. However, practical applications often require preserving the crease pattern. In this paper, we classify isometric deformations from a pillow box to a double rectangle among curved foldings that preserve the crease pattern. As a corollary, we prove that such an isometric deformation necessarily changes the topology of a pillow box.

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arXiv2024

Mannheim-d'Ocagne-Koenderink type formulas for asymptotic directions

We consider a surface embedded in the Euclidean 3-space and fix a tangential vector $v$ at a given point $p$ on the surface. In this paper, we first review a history of the formula obtained by Mannheim, d'Ocagne and Koenderink, which asserts that the Gaussian curvature of the surface at $p$ can be obtained if one knows "the normal curvature at $p$ with respect to $v$" and "the curvature of the contour line $L$ of the surface at $p$" with respect to the orthogonal projection induced by $v$. Unfortunately, this formula does not work when $v$ points in an asymptotic direction. When $v$ is just the case, we give anlogues of the formula, which include an invariant of cusp singular points of $L$.

math.DG↗
arXiv2024

Null hypersurfaces as wave fronts in Lorentz-Minkowski space

In this paper, we show that ``$L$-complete null hypersurfaces'' (i.e. ruled hypersurfaces foliated by entirety of light-like lines) as wave fronts in the $(n+1)$-dimensional Lorentz-Minkowski space are canonically induced by hypersurfaces in the $n$-dimensional Euclidean space. As an application, we show that most of null wave fronts can be realized as restrictions of certain $L$-complete null wave fronts. Moreover, we determine $L$-complete null wave fronts whose singular sets are compact.

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arXiv2024

Bour's theorem for helicoidal surfaces with singularities

In this paper, generalizing the techniques of Bour's theorem, we prove that every generic cuspidal edge, more generally, generic $n$-type edge, which is invariant under a helicoidal motion in Euclidean $3$-space admits non-trivial isometric deformations. As a corollary, several geometric invariants, such as the limiting normal curvature, the cusp-directional torsion, the higher order cuspidal curvature and the bias, are proved to be extrinsic invariants.

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arXiv2024

The total absolute curvature of closed curves with singularities

In this paper, we give a generalization of Fenchel's theorem for closed curves as frontals in Euclidean space $\mathbb{R}^n$. We prove that, for a non-co-orientable closed frontal in $\mathbb{R}^n$, its total absolute curvature is greater than or equal to $π$. It is equal to $π$ if and only if the curve is a planar locally $L$-convex closed frontal whose rotation index is $1/2$ or $-1/2$. Furthermore, if the equality holds and if every singular point is a cusp, then the number $N$ of cusps is an odd integer greater than or equal to $3$, and $N=3$ holds if and only if the curve is simple.

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arXiv2021

A generalization of Zakalyukin's lemma, and symmetries of surface singularities

Zakalyukin's lemma asserts that the coincidence of the images of two wave front germs implies the right equivalence of corresponding map germs under a certain genericity assumption. The purpose of this paper is to give an improvement of this lemma for frontals. Moreover, we give several applications for singularities on surfaces.

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arXiv2021

Symmetries of cross caps

It is well-known that cross caps on surfaces in the Euclidean 3-space can be expressed in Bruce-West's normal form, which is a special local coordinate system centered at the singular point. In this paper, we show a certain kind of uniqueness of such a coordinate system. In particular, the functions associated with this coordinate system produce new invariants on cross cap singular points. Using them, we classify the possible symmetries on cross caps.

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arXiv2021

Singularities of spacelike mean curvature one surfaces in de Sitter space

In this paper, we study the singularities of spacelike constant mean curvature one (CMC 1) surfaces in the de Sitter 3-space. We prove the duality between generalized conelike singular points and 5/2-cuspidal edges on spacelike CMC 1 surfaces. To describe the duality between $A_{k+3}$ singularities and cuspidal $S_k$ singularities, we introduce two invariants, called the $α$-invariant and $σ$-invariant, of spacelike CMC 1 surfaces at their singular points. Moreover, we give a classification of non-degenerate singular points on spacelike CMC 1 surfaces.

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arXiv2020

Geometry of lightlike locus on mixed type surfaces in Lorentz-Minkowski 3-space from a contact viewpoint

A surface in the Lorentz-Minkowski $3$-space is generally a mixed type surface, namely, it has the lightlike locus. We study local differential geometric properties of such a locus on a mixed type surface. We define a frame field along a lightlike locus, and using it, we define two lightlike ruled surfaces along a lightlike locus which can be regarded as lightlike approximations of the surface along the lightlike locus. We study a relationship of singularities of these lightlike surfaces and differential geometric properties of the lightlike locus. We also consider the intersection curve of two lightlike approximations, which gives a model curve of the lightlike locus.

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arXiv2020

Heinz-type mean curvature estimates in Lorentz-Minkowski space

We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space. As a corollary, we give a unified vanishing theorem of mean curvature for these entire graphs of constant mean curvature.

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arXiv2020

On the existence of four or more curved foldings with common creases and crease patterns

Consider an oriented curve $Γ$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding in the Euclidean space $\boldsymbol R^3$. This can be expressed as the image of an "origami map" $Φ:D\to \boldsymbol R^3$ such that $Γ$ is the singular set of $Φ$, the word "origami" coming from the Japanese term for paper folding. We call the singular set image $C:=Φ(Γ)$ the crease of $Φ$ and the singular set $Γ$ the crease pattern of $Φ$. We are interested in the number of origami maps whose creases and crease patterns are $C$ and $Γ$, respectively. Two such possibilities have been known. In the authors' previous work, two other new possibilities and an explicit example with four such non-congruent distinct curved foldings were established. In this paper, we determine the possibility of the number $N$ of congruence classes of curved foldings with the same crease and crease pattern. As a consequence, if $C$ is a non-closed simple arc, then $N=4$ if and only if both $Γ$ and $C$ do not admit any symmetries. On the other hand, when $C$ is a closed curve, there are infinitely many distinct possibilities for curved foldings with the same crease and crease pattern, in general.

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arXiv2020

Duality on generalized cuspidal edges preserving singular set images and first fundamental forms

In the second, fourth and fifth authors' previous work, a duality on generic real analytic cuspidal edges in the Euclidean 3-space $\boldsymbol R^3$ preserving their singular set images and first fundamental forms, was given. Here, we call this an `isometric duality'. When the singular set image has no symmetries and does not lie in a plane, the dual cuspidal edge is not congruent to the original one. In this paper, we show that this duality extends to generalized cuspidal edges in $\boldsymbol R^3$, including cuspidal cross caps, and $5/2$-cuspidal edges. Moreover, we give several new geometric insights on this duality.

math.DG↗
arXiv2020

Curved foldings with common creases and crease patterns

Consider a curve $Γ$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding $P$ in the Euclidean space $\boldsymbol R^3$. The singular set $C$ of $P$ as a space curve is called the crease of $P$ and the initially given plane curve $Γ$ is called the crease pattern of $P$. In this paper, we show that in general there are four distinct non-congruent curved foldings with a given pair consisting of a crease and crease pattern. Two of these possibilities were already known, but it seems that the other two possibilities (i.e. four possibilities in total) are presented here for the first time.

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arXiv2020

Cuspidal edges with the same first fundamental forms along a knot

Letting $C$ be a compact $C^ω$-curve embedded in $\boldsymbol R^3$ ($C^ω$ means real analyticity), we consider a $C^ω$-cuspidal edge $f$ along $C$. When $C$ is non-closed, in the authors' previous works, the local existence of three distinct cuspidal edges along $C$ whose first fundamental forms coincide with that of $f$ was shown, under a certain reasonable assumption on $f$. In this paper, if $C$ is closed, that is, $C$ is a knot, we show that there exist infinitely many cuspidal edges along $C$ having the same first fundamental form as that of $f$ such that their images are non-congruent to each other, in general.

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arXiv2020

Isometric deformations of wave fronts at non-degenerate singular points

Cuspidal edges and swallowtails are typical non-degenerate singular points on wave fronts in the Euclidean $3$-space. Their first fundamental forms belong to a class of positive semi-definite metrics called "Kossowski metrics". A point where a Kossowski metric is not positive definite is called a singular point or a semi-definite point of the metric. Kossowski proved that real analytic Kossowski metric germs at their non-parabolic singular points(the definition of "non-parabolic singular point" is stated in the introduction here) can be realized as wave front germs (Kossowski's realization theorem). On the other hand, in a previous work with K. Saji, the third and the fourth authors introduced the notion of "coherent tangent bundle". Moreover, the authors, with M. Hasegawa and K. Saji, proved that a Kossowski metric canonically induces an associated coherent tangent bundle. In this paper, we shall explain Kossowski's realization theorem from the viewpoint of coherent tangent bundles. Moreover, as refinements of it, we give a criterion that a given Kossowski metric can be realized as the induced metric of a germ of cuspidal edge (resp. swallowtail or cuspidal cross cap). Several applications of these criteria are given. Also, some remaining problems on isometric deformations of singularities of analytic maps are given at the end of this paper.

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arXiv2020

Behavior of torsion functions of spacelike curves in Lorentz-Minkowski space

In this paper, we introduce the pseudo-torsion functions along spacelike curves whose curvature vector field has isolated lightlike points in Lorentz-Minkowski 3-space, and prove the fundamental theorem. Moreover, we analyze the behavior of the torsion function at such points. As a corollary, we obtain a necessary and sufficient condition for real analytic spacelike curves to be planar.

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arXiv2019

Mixed type surfaces with bounded Gaussian curvature in three-dimensional Lorentzian manifolds

A mixed type surface is a connected regular surface in a Lorentzian 3-manifold with non-empty spacelike and timelike point sets. The induced metric of a mixed type surface is a signature-changing metric, and their lightlike points may be regarded as singular points of such metrics. In this paper, we investigate the behavior of Gaussian curvature at a non-degenerate lightlike point of a mixed type surface. To characterize the boundedness of Gaussian curvature at a non-degenerate lightlike points, we introduce several fundamental invariants along non-degenerate lightlike points, such as the lightlike singular curvature and the lightlike normal curvature. Moreover, using the results by Pelletier and Steller, we obtain the Gauss-Bonnet type formula for mixed type surfaces with bounded Gaussian curvature.

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