Envelope singularities of families of planes in control theory

Zakalyukin, V. M.; Kurbatskii, A. N.

doi:10.1134/s0081543808030073

Cited by 22 publications

(42 citation statements)

References 3 publications

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“…We remark that the germs of types B 2 , B 3 , C 3 , B 4 , C 4 , F 4 appeared in the classification of big fronts in [27]. However, these germs cannot be realized as graphlike generating families.…”

Section: The Cases N =mentioning

confidence: 89%

“…Almost all results are given by Zakalyukin [27]. However, we give some detailed information here which might be new.…”

Section: Appendix B Families Of Wave Fronts and Discriminantmentioning

confidence: 94%

“…By Theorem 9.2, there exists an open dense subset O ⊂ Emb s (U, LC * ) such that for any x + ∈ O, the germ of the graphlike Legendrian unfolding L H X at each point is S.P + -Legendrian stable, where X = T, S. By the local classification theorem on graphlike Legendrian unfoldings by the S.P + -Legendrian equivalence [10,27], the corresponding graphlike Legendrian unfolding germ L H X (X = T, S) at any point is S.P + -Legendrian equivalent to a graphlike Legendrian unfoldings whose graphlike generating family is stably x-S.P + -K-equivalent to one of the following germs:…”

Section: The Cases N =mentioning

confidence: 99%

“…However, we give some detailed information here which might be new. Moreover some equivalence relations presented here have been independently introduced by the first named author [10] for different purposes from those of Zakalyukin [27].…”

Section: Appendix B Families Of Wave Fronts and Discriminantmentioning

confidence: 99%

“…We study generic properties in §9. In §10 we apply the classification results in [10,27] to the case for n = 3 and draw some pictures.…”

Section: Introductionmentioning

confidence: 99%

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Spacelike parallels and evolutes in Minkowski pseudo-spheres

Izumiya

Takahashi

2007

Journal of Geometry and Physics

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We consider extrinsic differential geometry on spacelike hypersurfaces in Minkowski pseudo-spheres (hyperbolic space, de Sitter space and the lightcone). In the previous paper [18] we have shown a basic Legendrian duality theorem between pseudo-spheres. We define the spacelike parallels by using the basic Legendrian duality theorem. This definition unifies the notions of parallels of spacelike hypersurfaces in pseudo-spheres. We also define the evolute as the locus of singularities of the spacelike parallels. These notions are investigated as applications of Lagrangian or Legendrian singularity theory. We consider geometric properties of non-singular spacelike hypersurfaces corresponding to singularities of spacelike parallels or evolutes.

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Section: The Cases N =mentioning

confidence: 89%

“…Almost all results are given by Zakalyukin [27]. However, we give some detailed information here which might be new.…”

Section: Appendix B Families Of Wave Fronts and Discriminantmentioning

confidence: 94%

Section: The Cases N =mentioning

confidence: 99%

Section: Appendix B Families Of Wave Fronts and Discriminantmentioning

confidence: 99%

“…We study generic properties in §9. In §10 we apply the classification results in [10,27] to the case for n = 3 and draw some pictures.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Spacelike parallels and evolutes in Minkowski pseudo-spheres

Izumiya

Takahashi

2007

Journal of Geometry and Physics

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Singular Points of the Wigner Caustic and Affine Equidistants of Planar Curves

Domitrz

Zwierzyński

2019

Bull Braz Math Soc, New Series

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The Gauss–Bonnet Theorem for Coherent Tangent Bundles over Surfaces with Boundary and Its Applications

Domitrz¹,

Zwierzyński²

2019

J Geom Anal

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In [34,35,36] the Gauss-Bonnet formulas for coherent tangent bundles over compact oriented surfaces (without boundary) were proved. We establish the Gauss-Bonnet theorem for coherent tangent bundles over compact oriented surfaces with boundary. We apply this theorem to investigate global properties of maps between surfaces with boundary. As a corollary of our results we obtain Fukuda-Ishikawa's theorem. We also study geometry of the affine extended wave fronts for planar closed non singular hedgehogs (rosettes). In particular, we find a link between the total geodesic curvature on the boundary and the total singular curvature of the affine extended wave front, which leads to a relation of integrals of functions of the width of a resette.2010 Mathematics Subject Classification. Primary 57R45, Secondary 53A05.

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Envelope singularities of families of planes in control theory

Cited by 22 publications

References 3 publications

Spacelike parallels and evolutes in Minkowski pseudo-spheres

Spacelike parallels and evolutes in Minkowski pseudo-spheres

Singular Points of the Wigner Caustic and Affine Equidistants of Planar Curves

The Gauss–Bonnet Theorem for Coherent Tangent Bundles over Surfaces with Boundary and Its Applications

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