@article{Alharbi2025, 
author = {Fawaz Alharbi},
title = {Height–plane contact along edges, corners, and cusps},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {12},
pages = {30594-30622},
keywords = {Bifurcation diagram, caustic, vector field, cuspidal edge, contact, curvatures and torsions, height function, deformations, discriminant},
url = {https://www.sciopen.com/article/10.3934/math.20251341},
doi = {10.3934/math.20251341},
abstract = {We study pairs              X      ~        =  (  X  ,  S  ) which consist of regular surfaces    X  ⊂                    R              3   endowed with a distinguished boundary    S given by        y    2    −      x    s    =  0 (   s  =  1  ,  2  ,  3: edge, corner, cusp). First, we give an explicit description of the logarithmic vector fields tangent to    (  g  ,      b    s    ),    g  =  z  −  f  (  x  ,  y  ). In particular, the five fields    E  ,  L  ,      G    g    ,      G    x    ,      G    y   generate        D    e    r    (  −  log  ⁡  (  g  ,      b    s    )  ). This yields a concrete Kodaira–Spencer calculus and a relative    2-determinacy theorem (with a single parabolic exception). Then, we classify submersions and obtain normal forms and mini versal unfoldings for submersion germs in codimension    ≤  2, with respect to the        R    (            X      ~        )-equivalence relation. Second, for height functions        h    v    (  w  )  =  ⟨  w  ,  v  ⟩ we obtain sharp linear conditions on    v  ∈      S    2   that characterize the        A    k  –contact along each boundary type. In particular, in the cuspidal case, the        A    3        /        A    5   transition is governed by singular torsion. The resulting direction discriminant              D        s    ⊂      S    2   corresponds, under the direction–parameter normalization, to a hyperplane arrangement of linear discriminant loci in the mini versal parameter spaces. Both the ambient discriminant              D        X   and the boundary contact discriminants are invariant under    P–             R              +        (            X      ~        )–equivalence.}
}