266 research outputs found
Ruled Laguerre minimal surfaces
A Laguerre minimal surface is an immersed surface in the Euclidean space
being an extremal of the functional \int (H^2/K - 1) dA. In the present paper,
we prove that the only ruled Laguerre minimal surfaces are up to isometry the
surfaces R(u,v) = (Au, Bu, Cu + D cos 2u) + v (sin u, cos u, 0), where A, B, C,
D are fixed real numbers. To achieve invariance under Laguerre transformations,
we also derive all Laguerre minimal surfaces that are enveloped by a family of
cones. The methodology is based on the isotropic model of Laguerre geometry. In
this model a Laguerre minimal surface enveloped by a family of cones
corresponds to a graph of a biharmonic function carrying a family of isotropic
circles. We classify such functions by showing that the top view of the family
of circles is a pencil.Comment: 28 pages, 9 figures. Minor correction: missed assumption (*) added to
Propositions 1-2 and Theorem 2, missed case (nested circles having nonempty
envelope) added in the proof of Pencil Theorem 4, missed proof that the arcs
cut off by the envelope are disjoint added in the proof of Lemma
Anisotropic Wavefronts and Laguerre Geometry
Motivated by the study of wave fronts in anisotropic media, we propose an
incidence geometry of anisotropic spheres in a Finsler-Minkowski space. An
anisotropic version of the Laguerre functional is considered. In some
circumstances, this functional can be used to determine that two wavefronts
observed at distinct times in a homogeneous, anisotropic medium, do not
originate from the same source
Classification results on surfaces in the isotropic 3-space
The isotropic 3-space I^3 which is one of the Cayley--Klein spaces is
obtained from the Euclidean space by substituting the usual Euclidean distance
with the isotropic distance. In the present paper, we give several
classifications on the surfaces in I^3 with the constant relative curvature
(analogue of the Gaussian curvature) and the constant isotropic mean curvature.
In particular, we classify the helicoidal surfaces in I^3 with constant
curvature and analyze some special curves on these.Comment: 12 pages, 2 figure
3D stochastic bicontinuous microstructures: Generation, topology and elasticity
Motivated by recent experimental investigations of the mechanical behavior of nanoporous metal we explore an efficient and robust method for generating 3D representative volume elements (RVEs) with strikingly similar behavior. Our approach adopts Cahn's method of generating a Gaussian random field by taking a superposition of standing sinusoidal waves of fixed wavelength but random in direction and phase. In its theory part, our study describes closed-form expressions for how the solid volume fraction affects the binarization level, mean structure size, specific surface area, averages of mean and Gaussian curvature, and the scaled topological genus. Based on numerical studies we report on criteria for achieving representative realizations of the structure by proper choice of the number of waves and element size. We also show that periodic structures are readily created. We analyze the mechanical properties considering linear and infinitesimal elasticity and evaluate the residual anisotropy (which can be made small) and the effective values of the Young's modulus and Poisson's ratio. The numerical results are in excellent agreement with experimental findings for the variation of stiffness with solid fraction of nanoporous gold made by dealloying. We propose scaling relations that achieve naturally a perfect agreement with the numerical and experimental data. The scaling relation for the stiffness accounts for a percolation-to-cluster transition in the random field microstructure at a finite solid fraction. We propose that this transition is the origin of the previously reported anomalous compliance of nanoporous gold
Spinor representation in isotropic 3-space via Laguerre geometry
We give a detailed description of the geometry of isotropic space, in
parallel to those of Euclidean space within the realm of Laguerre geometry.
After developing basic surface theory in isotropic space, we define spin
transformations, directly leading to the spinor representation of conformal
surfaces in isotropic space. As an application, we obtain the Weierstrass-type
representation for zero mean curvature surfaces, and the Kenmotsu-type
representation for constant mean curvature surfaces, allowing us to construct
many explicit examples.Comment: 30 pages, 9 figure
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