Circular surfaces

Izumiya, Shyūichi; Saji, Kentaro; Takeuchi, Nobuko

doi:10.1515/advgeom.2007.017

Cited by 43 publications

(33 citation statements)

References 21 publications

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“…The striction curve is a curve on the ruled surface which contains the singularities of the surface. Similarly, an analogous notion of the striction curve also plays a crucial role for one-parameter families of circles ( [23] [23]. We shall investigate these surfaces in the forthcoming paper.…”

mentioning

confidence: 99%

The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and "flat" spacelike surfaces

Izumiya¹,

Saji²

2010

J. Sing.

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Using the Legnedrian duarities between surfaces in pseudo-spheres in Lorentz-Minkowski 4-space, we study various kind of flat surfaces in pseudo-spheres. We consider a surface in the pseudo-sphere and its dual surface. Flatness of a surface is defined by the degeneracy of the dual surface similar to the case for the Gauss map of a flat surface in the Euclidean space. We study singularities of these flat surfaces and dualities of singularities.

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mentioning

confidence: 99%

The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and "flat" spacelike surfaces

Izumiya¹,

Saji²

2010

J. Sing.

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“…We can recognize the type of a smooth curve germ by using the following simple calculations. Let 7 : / -> R 3 be a space like curve with type A at to-Then we can show the following assertions ( [13,9,10,5,7]). …”

Section: Ln-mmentioning

confidence: 99%

“…Since the Gauss curvature is the determinant of the differential of the Gauss map, S = x(U) is a developable surface if and only if the Gauss map of the surface is singular at any point of S. It has been known that a developable surface is a ruled surface [14]. A ruled surface in M 3 is a surface given by a one-parameter family of lines [7,14]. It is locally defined as a mapping : / x R -> R 3 by F^S)(t,u) = 7(t)+uS(t), where 7 : I -> R 3 , We remark that once we have the above classification theorem, the notion of the developable surfaces is independent of the metric structure of R 3 .…”

Section: Ln-mmentioning

confidence: 99%

Lightlike Developables in Minkowski 3-Space

Chino¹,

Izumiya²

2010

Demonstratio Mathematica

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Abstract. We say that a surface in Minkowski 3-space is a lightlike developable if all pseudo-normal vectors of the regular part of the surface sire lightlike. The tangent surface of a lightlike curve is one of the lightlike developables. We give a generic classification of such surfaces. The all arguments in this paper are elementary. However, we discovered the H3 type singularity appears in generic for such a class of surfaces. Since the H3 type singularity usually appears in non-generic situation, this is a quite interesting phenomenon. IntroductionA surface in Euclidean space whose Gauss curvature vanishes on the regular part is called a developable surface. It has been known that a developable surface is a part of a conical surface, a cylindrical surface, the tangent surface of a space curve or the glue of such surfaces. Developable surfaces have singularities in general. The tangent surface of a space curve has the most interesting singularities in the above three kinds of surfaces. In fact Cleave [2] showed that the germ of the tangent surface of a generic 2000 Mathematics Subject Classification: 53A35, 58C27, 58C28.

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“…Since the Gauss curvature is the determinant of the differential of the Gauss map, S = x(U ) is a developable surface if and only if the Gauss map of the surface is singular at any point of S. It has been known that a developable surface is a ruled surface [14]. A ruled surface in R 3 is a surface given by a one-parameter family of lines [7,14]. It is locally defined as a mapping F (γ,δ) : I ×R → R 3 by F (γ,δ) (t, u) = γ(t)+uδ(t), where γ : I → R 3 , δ : I → R 3 \ {0} are smooth mappings.…”

Section: Developable Surfaces In Euclidean Spacementioning

confidence: 99%

Lightlike developables in Minkowski 3-space

Chino¹,

Izumiya²

2010

Demonstratio Mathematica

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Abstract. We say that a surface in Minkowski 3-space is a lightlike developable if all pseudo-normal vectors of the regular part of the surface are lightlike. The tangent surface of a lightlike curve is one of the lightlike developables. We give a generic classification of such surfaces. The all arguments in this paper are elementary. However, we discovered the H 3 type singularity appears in generic for such a class of surfaces. Since the H 3 type singularity usually appears in non-generic situation, this is a quite interesting phenomenon. IntroductionA surface in Euclidean space whose Gauss curvature vanishes on the regular part is called a developable surface. It has been known that a developable surface is a part of a conical surface, a cylindrical surface, the tangent surface of a space curve or the glue of such surfaces. Developable surfaces have singularities in general. The tangent surface of a space curve has the most interesting singularities in the above three kinds of surfaces. In fact Cleave [2] showed that the germ of the tangent surface of a generic cuspdialedge cuspidal cross cap 2000 Mathematics Subject Classification: 53A35, 58C27, 58C28.

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Circular surfaces

Abstract: A circular surface is a one-parameter family of standard circles in R 3 . In this paper some correspondences between the properties of circular surfaces and those of classical ruled surfaces are investigated. Singularities of circular surfaces are also studied.

Cited by 43 publications

References 21 publications

The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and "flat" spacelike surfaces

The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and "flat" spacelike surfaces

Lightlike Developables in Minkowski 3-Space

Lightlike developables in Minkowski 3-space

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