Recognition Problem of Frontal Singularities

Ishikawa, Goo

doi:10.5427/jsing.2020.21i

Cited by 9 publications

(3 citation statements)

References 17 publications

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“…We have that the Jacobian ideal of Tan(f ) is principal and, in fact, is generated by st a 2 −a 1 −1 . Therefore Tan(f ) has a nowhere dense singular locus and it turns out to be a proper frontal (see [15], Lemma 2.3).…”

Section: Parallels To Tangent Surfaces Of Frontal Curvesmentioning

confidence: 98%

Singularities of Developable Surfaces

Ishikawa

1999

Singularity Theory

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It is known that the class of developable surfaces which have zero Gaussian curvature in three dimensional Euclidean space is preserved by the parallel transformations. A tangent developable surface is defined as a ruled developable surface by tangent lines to a space curve and it has singularities at least along the space curve, called the directrix or the the edge of regression. Also the class of tangent developable surfaces are invariant under the parallel deformations. In this paper the notions of tangent developable surfaces and their parallels are naturally generalized for frontal curves in general in Euclidean spaces of arbitrary dimensions. We study singularities appearing on parallels to tangent developable surfaces of frontal curves and give the classification of generic singularities on them for frontal curves in 3 or 4 dimensional Euclidean spaces.

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Section: Parallels To Tangent Surfaces Of Frontal Curvesmentioning

confidence: 98%

Singularities of Developable Surfaces

Ishikawa

1999

Singularity Theory

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“…In recent years, there is a great interest in the geometry of a special type of singular surface, namely, frontal. Many papers are dedicated to the study of frontals from singularity theory and geometry viewpoints [6, 7, 12], in particular wave fronts a subclass of these [1, 9, 11, 13]. The word “front” comes from physical fronts, bounding a domain in which a physical process propagates at a fixed moment in time.…”

Section: Introductionmentioning

confidence: 99%

“…The cuspidal edge and the swallowtail (see Figures 1 and 2) are two types of singular points that represent the generic singularities in the space of wave fronts with the Whitney

C^{\infty }

‐topology. For this reason, all the re‐parametrizations and diffeomorphic singular surfaces to these are the most studied and there exist criterias to recognize them [7, 8]. However, these singularities are not generic in the space of all frontals (in fact proper frontals are not generic either) [6].…”

Section: Introductionmentioning

confidence: 99%

The fundamental theorem for singular surfaces with limiting tangent planes

2022

Mathematische Nachrichten

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In this paper, we prove a similar result to the fundamental theorem of regular surfaces in classical differential geometry, which extends the classical theorem to the entire class of singular surfaces in Euclidean 3-space known as frontals.Also, we characterize in a simple way these singular surfaces and its fundamental forms with local properties in the differential of its parametrization and decompositions in the matrices associated to the fundamental forms. In particular, we introduce new types of curvatures that can be used to characterize wave fronts.The only restriction on the parametrizations that is assumed in several occasions is that the singular set has empty interior.

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The axial curvature for corank 1 singular surfaces

Sinha¹,

Saji²

2022

Tohoku Math. J. (2)

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Recognition Problem of Frontal Singularities

Cited by 9 publications

References 17 publications

Singularities of Developable Surfaces

Singularities of Developable Surfaces

The fundamental theorem for singular surfaces with limiting tangent planes

The axial curvature for corank 1 singular surfaces

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