Singularities of affine equidistants: projections and contacts

Domitrz, Wojciech; Rios, Pedro de M.; Ruas, María Aparecida Soares

doi:10.5427/jsing.2014.10d

Cited by 10 publications

(27 citation statements)

References 13 publications

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“…The singularities and the geometry of affine λ-equidistants were very widely studied in many papers [1,5,6,8,13,17,33,40]. The envelope of affine diameters (the Centre Symmetry Set) was studied in [7,12,14,15,16].…”

Section: Geometry Of the Affine Extended Wave Frontmentioning

confidence: 99%

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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Section: Geometry Of the Affine Extended Wave Frontmentioning

confidence: 99%

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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“…The affine λ-equidistant is the set of points divided chords connecting points on M where tangent lines to M are parallel in the ratio λ. There are many papers considering affine equidistants, see [4,6,7,8,9,14,16,24,32,36]. In [4,14] the Wigner caustic is known as the area evolute and in [24] is known as the symmetry defect.…”

Section: Introductionmentioning

confidence: 99%

Isoperimetric equalities for rosettes

Zwierzyński

2020

Int. J. Math.

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In this paper we study the isoperimetric-type equalities for rosettes, i.e. regular closed planar curves with non-vanishing curvature. We find the exact relations between the length and the oriented area of rosettes based on the oriented areas of the Wigner caustic, the Constant Width Measure Set and the Spherical Measure Set.We also study and find new results about the geometry of affine equidistants of rosettes and of the union of rosettes.2010 Mathematics Subject Classification. Primary: 52A38, 53A04, 58K70. Secondary: 52A40.

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“…It leads to the construction of bi-dimensional improper affine spheres ( [3]). The Wigner caustic is an example of an affine λ-equidistant, which is the locus of points dividing chords connecting points on M with parallel tangent lines in a fixed ratio λ ( [5,8,10,23]).…”

Section: Introductionmentioning

confidence: 99%