Structure and transitions of line bitangencies in a family of surface pairs

Dreibelbis, Daniel; Olsen, W.

doi:10.1007/s00022-022-00649-z

Cited by 2 publications

(3 citation statements)

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“…See §7. The criterion (7) above for a two real branches coincides with the condition c 3 < 0 for a beaks transition in §7. The criminant is locally diffeomorphic to a folded Whitney umbrella.…”

Section: Geometrical Propertiessupporting

confidence: 59%

“…In an unpublished Master's thesis [20] (see also [7]), W.E.Olsen, a student of Daniel Dreibelbis, studies bitangent lines and planes to two surface patches M, N by considering the corresponding subsets of M × N and finds geometrical conditions for singularities of their projections to the two factors. He also gives examples of 1-parameter deformations of the surfaces and considers the more degenerate case of double-points, that is points of intersection of M and N .…”

Section: Other Work In This Areamentioning

confidence: 99%

“…The binodal curves on both M and N undergo a lips transition if c 3 > 0 and a beaks transition if c 3 < 0. (See (7) in §5 for a simplified version of c 3 < 0 when special coordinates are used. )…”

mentioning

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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Section: Geometrical Propertiessupporting

confidence: 59%

Section: Other Work In This Areamentioning

confidence: 99%