Equidistants and their duals for families of plane curves

Giblin, Peter; Reeve, Graham

doi:10.2969/aspm/07810251

Cited by 2 publications

(3 citation statements)

References 7 publications

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“…Since our investigation is local we shall in fact consider two surface patches M 0 and N 0 which vary in a 1-parameter family M ε , N ε . A similar degeneracy was investigated for plane curves in [6]; we sometimes call it a 'supercaustic' situation. This term is defined in §2.…”

Section: Introductionmentioning

confidence: 71%

“…This means that, for all λ, but ε = 0, the critical set itself is singular at the origin of R 4 . Definition 2.3 In the above situation, the λ-axis is called a supercaustic; see [6]. The whole of this axis maps to singular points of the equidistants.…”

Section: Maps and Supercausticsmentioning

confidence: 99%

“…The main results are contained in Proposition 3.2 and the accompanying Figure 1 for Generic Case 1.1; Proposition 3.4 and the accompanying Figure 4 for Special Case 1.2, and Table 1 in §4. 6 for Degenerate Case 2.…”

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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Section: Introductionmentioning

confidence: 71%

Section: Maps and Supercausticsmentioning

confidence: 99%

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Equidistants for families of surfaces

Giblin¹,

Reeve²

2020

jsing

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The λ-Point Map between Two Legendre Plane Curves

Alghanemi

AlGhawazi

2023

Mathematics

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The λ-point map between two Legendre plane curves, which is a map from the plane into the plane, is introduced. The singularity of this map is studied through this paper and many known plane map singularities are realized as special cases of this construction. Precisely, the corank one and corank two singularities of the λ-point map between two Legendre plane curves are investigated and the geometric conditions for this map to have corank one singularities, such as fold, cusp, swallowtail, lips, and beaks are obtained. Additionally, the geometric conditions for the λ-point map to have a sharksfin singularity, which is a corank two singularity, are obtained.

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Equidistants and their duals for families of plane curves

Abstract: In this paper a Minkowski analogue of the Euclidean medial axis of a closed and smooth plane curve is introduced. Its generic local configurations are studied and the types of shocks that can occur on it are determined.

Cited by 2 publications

References 7 publications

Equidistants for families of surfaces

Equidistants for families of surfaces

The λ-Point Map between Two Legendre Plane Curves

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