Singularities of the affine chord envelope for two surfaces in four-space

Reeve, Graham; Zakalyukin, V. M.

doi:10.1134/s0081543812040165

Cited by 3 publications

(4 citation statements)

References 9 publications

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“…Consider the space W of function germs of the type F = −q 3 +εH(u, v, ε, q 1 , q 2 , τ ). Following [10,21] we show stability of the generating function inside this space W .…”

Section: Expanding the Generating Functionmentioning

confidence: 92%

“…Following the method used in [11, p.44-45], [23], we use the generating family, for each fixed τ near 0,…”

Section: Generating Familiesmentioning

confidence: 99%

“…In part we are responding to a suggestion on page 85 of [25]: 'It would be interesting to compare Korteweg's method of continuous deformation of surfaces with the methodology of catastrophe theory'. But the major part of our mathematical contribution is to give a complete list of normal forms of singularities up to codimension 1 (generic 1-parameter families of surfaces) occurring in the bitangent plane context, extending the lists in [10] and [23].…”

Section: Introductionmentioning

confidence: 99%

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Equidistants for families of surfaces

Giblin¹,

Reeve²

2020

jsing

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For a smooth surface in R 3 this article contains local study of certain affine equidistants, that is loci of points at a fixed ratio between points of contact of parallel tangent planes (but excluding ratios 0 and 1 where the equidistant contains one or other point of contact). The situation studied occurs generically in a 1-parameter family, where two parabolic points of the surface have parallel tangent planes at which the unique asymptotic directions are also parallel. The singularities are classified by regarding the equidistants as critical values of a 2-parameter unfolding of maps from R 4 to R 3 . In particular, the singularities that occur near the so-called 'supercaustic chord', joining the two special parabolic points, are classified. For a given ratio along this chord either one or three special points are identified at which singularities of the equidistant become more special. Many of the resulting singularities have occurred before in the literature in abstract classifications, so the article also provides a natural geometric setting for these singularities, relating back to the geometry of the surfaces from which they are derived. MR Classification 57R45, 53A05

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“…Consider the space W of function germs of the type F = −q 3 +εH(u, v, ε, q 1 , q 2 , τ ). Following [10,21] we show stability of the generating function inside this space W .…”

Section: Expanding the Generating Functionmentioning

confidence: 92%

“…Following the method used in [11, p.44-45], [23], we use the generating family, for each fixed τ near 0,…”

Section: Generating Familiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Equidistants for families of surfaces

Giblin¹,

Reeve²

2020

jsing

View full text Add to dashboard Cite

show abstract

“…It was studied by many authors, also in higher dimensions, and in Minkowski spaces. For instance, authors in [19,20,21,25] analyse the local properties of the CSS based on the theory of Lagrange and Legendre singularities. In [17,18] the authors study the Centre Symmetry Set of families of plane curves and of families of surfaces.…”

Section: Introductionmentioning

confidence: 99%