Umbilic points on Gaussian random surfaces

Abstract

An umbilic point U on a surface Z is a place where the two principal curvatures of Z are equal. U is a Singularity of Z in three different senses: (i) it is the source of elliptic (E) or hyperbolic (H) umbilic catastrophes in the envelope of normals ('focal surface') of Z; (ii) it has index *; depending on whether the principal curvature directions of Z (defining the lines of curvature) rotate by f ?~ during a circuit of U; (iii) it has a pattern of the 'star' (S), 'lemon' (L) or 'monstar' (M) type depending on the configuration of lines of curvature near U. We calculate the average density of umbilic points, and the fractions a of umbilin of the different sorts, for the case where X is a surface whose (small) deviation from a plane is specified by a Gaussian random function. It is always the case that as = = f. The other fractions depend on the degree of isotropy of the statistics of Z. In the isotropic case the elliptic umbilic fraction is aE = 1-aH = 0.268, and the monstar fraction is a M = ;-a L = 0. 0 5 3 .

Local phase structure of wave dislocation lines: twist and twirl

“…After some analysis, however, it can be shown that there is no unique phase contour whose local transverse curvature vanishes: generically, there are either one or three phase contours with this property. The number of phase contours in question is given by the number of real roots of a certain cubic in tan φ, reminiscent of the number of straight lines terminating on a line field singularity (lemon versus star and monstar) [28,21]. An example of a solution to the wave equation with three phase lines of vanishing transverse curvature is ψ = x + iy + ik(x 2 − y 2 /2)/2 exp(ikz).…”

A metric property of umbilic points

“…This paper is the outcome of an initial attempt to provide a mathematical formulation and a proof, in the tradition of Geometry and Classical Analysis, that could correspond to the conclusions of (Berry and Hannay 1977), reported in the tradition of Statistical Physics.…”

“…In the domain of Statistical Physics, but still connected to Geometry and Topology, Berry and Hannay (Berry and Hannay 1977) carried out a quantitative statistical study of the proportions in which the different types of umbilic types are distributed in random surfaces, such as those modeling an ocean or a lake. An issue here is to study how the presence of umbilic points in a random surface influences the reflection on it of electromagnetic short waves.…”

“…Considering only the local aspect of surfaces at umbilic points, this discrepancy may be due to the fact that in the calculations made in (Berry and Hannay 1977), the cubic forms are regarded…”

Generic singularities of line fields on 2D manifolds

“…Three types of singularities, called Lemon, Monstar and Star (see [2]), were identified by Darboux in [7] (see Figure 2). It was proven in [19] that Lemon, Monstar and Star are the structurally stable singularities of lines of principal curvature with respect to the Whitney C 3 -topology of immersions of a surface in R 3 .…”