Umbilic points on Gaussian random surfaces

@article{Berry1977UmbilicPO,
  title={Umbilic points on Gaussian random surfaces},
  author={Michael V. Berry and J H Hannay},
  journal={Journal of Physics A},
  year={1977},
  volume={10},
  pages={1809-1821}
}
An umbilic point U on a surface Sigma is a place where the two principal curvatures of Sigma are equal. U is a singularity of Sigma 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 Sigma ; (ii) it has index +or-1/2 depending on whether the principal curvature directions of Sigma (defining the lines of curvature) rotate by +or- pi during a circuit of U; (iii) it has a pattern of the 'star' (S… 

Figures from this paper

Principal Foliations of Surfaces near Ellipsoids

The lines of curvature of a surface embedded in $\R^3$ comprise its principal foliations. Principal foliations of surfaces embedded in $\R^3$ resemble phase portraits of two dimensional vector

Generic curvature features from 3-D images

The author presents a method for the computation of certain singularities-parabolic and umbilic points-of the principal curvature and direction fields of surfaces in three-dimensional images, and develops a refinement algorithm to that end on the basis of an iterative minimization over the estimated local structure at surface trace points.

Optical caustics from liquid drops under gravity: observations of the parabolic and symbolic umbilics

  • J. Nye
  • Physics
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1979
An earlier paper (Nye 1978) discussed the caustics formed by refraction of light in a thin gravity-free water drop resting on a glass surface. Here the effect of gravity is examined. In an irregular

A metric property of umbilic points

In the space of cubic forms of surfaces, regarded as a G-space and endowed with a natural invariant metric, the ratio of umbilic points with a negative index to those with a positive index is

Monstars on Glaciers

  • J. Nye
  • Geology
    Journal of Glaciology
  • 1983
Abstract Isotropic points are structurally stable features of any complicated field of stress or strain-rate, and therefore will almost always be present on the surface of a glacier. A given

Experimental optical diabolos.

This work presents measured contour maps, surfaces, cones, and diabolos of a and b for a random ellipse field (speckle pattern) and presents the profound effects of the singularity in alpha on the orientation of the ellipses surrounding the C point.

Umbilic lines in orientational order

Three-dimensional orientational order in systems whose ground states possess nonzero gradients typically exhibits linelike structures or defects: λ lines in cholesterics or Skyrmion tubes in
...

References

SHOWING 1-6 OF 6 REFERENCES

The statistical analysis of a random, moving surface

  • M. Longuet-Higgins
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1957
The following statistical properties are derived for a random, moving, Gaussian surface: (1) the probability distribution of the surface elevation and of the magnitude and orientation of the

The normal singularities of a submanifold

Let R be furnished with its Euclidean bilinear scalar product R x R n -H> R; (JC, JCO H^ χ> x and associated positive-definite quadratic form R -H> R; x !-• χ = X'X, and let M be an ra-dimensional

Waves and Thom's theorem

Abstract Short-wave fields can be well approximated by families of trajectories. These families are dominated by their singularities, i.e. by caustics, where the density of trajectories is infinite.

Structural stability and morphogenesis

  • R. Thom
  • Computer Science, Physics
    Pattern Recognit.
  • 1976

Photoelusticity (London: Cleaver-Hume) Longuet-Higgins

  • M S 1956 Phil. Trans. R. Soc. A
  • 1949