CMC hypersurfaces with two principal curvatures

We describe the geometric structures of the lines of principal curvature and the partially umbilic singularities of the three-dimensional non compact generic quadric hypersurfaces of R4. This includes the ellipsoidal hyperboloids of one and two sheets and the toroidal hyperboloids. The present study complements the analysis of the compact ellipsoidal hypersurfaces carried out in [9].

Constructing PDE-based surfaces bounded by geodesics or lines of curvature

Chaotic mixing in a closed vessel is studied experimentally and numerically in different two-dimensional (2D) flow configurations. For a purely hyperbolic phase space, it is well known that concentration fluctuations converge to an eigenmode of the advection-diffusion operator and decay exponentially with time. We illustrate how the unstable manifold of hyperbolic periodic points dominates the resulting persistent pattern. We show for different physical viscous flows that, in the case of a fully chaotic Poincaré section, parabolic periodic points at the walls lead to slower (algebraic) decay. A persistent pattern, the backbone of which is the unstable manifold of parabolic points, can be observed. However, slow stretching at the wall forbids the rapid propagation of stretched filaments throughout the whole domain, and hence delays the formation of an eigenmode until it is no longer experimentally observable. Inspired by the baker's map, we introduce a 1D model with a parabolic point that gives a good account of the slow decay observed in experiments. We derive a universal decay law for such systems parametrized by the rate at which a particle approaches the no-slip wall.

Perception of three-dimensional shape from texture is based on patterns of oriented energy

Texture classification algorithms using local binary pattern (LBP) and its variants usually can achieve attractive results. However, the selected rotation invariant structural patterns in numerous LBP variants are not absolutely continuous invariant to any rotation angle. To improve the classification effectiveness on this occasion, in this paper, we introduce a robust descriptor based on the principal curvatures (PCs) and rotation invariant version of CLBP_Sign operator in completed LBP (CLBP), namely PC-LBP. Different from the original LBP and many LBP variants, PCs are employed in this paper to represent each local structure information due to their continuous rotation invariance. Simultaneously, both micro-and macro-structure texture information can also be captured through PCs, which comprise maximum and minimum curvatures. Inspired by the similar coding strategy of the CLBP_Sign operator, a new operator CLBP_PC is developed. By exploiting complementary information resulting from the two operators combination, the final PC-LBP descriptor has the properties of conspicuous rotation invariance, strong discriminativeness, gray scale invariance, needless of pretraining, and high computational efficiency. In addition, to improve the robustness of texture classification with multiresolution, a multiscale sampling approach is designed by adjusting three parameters accordingly. Experimental results demonstrate that the proposed multiresolution PC-LBP approach achieves comparable performance or outperforms a large number of state-of-the-art methods. Impressively, the classification accuracy of the proposed method performed on Outex_TC_00010 test suite is 100%.

Lines of axial curvature on surfaces immersed in [formula omitted

Axis estimation and grouping of rotationally symmetric object segments

The Richtmyer-Meshkov instability (RMI) of single-mode air-SF(6) interfaces is studied numerically and the emphasis is placed on the effect of the principal curvature on the early evolution of the shocked interface. Two three-dimensional initial interfaces with opposite (3D-) and identical (3D+) principal curvatures and a traditional two-dimensional interface (2D) are considered. The weighted essentially nonoscillatory scheme and the Level-Set method combined with the real ghost fluid method are adopted. For comparison, perturbations on the initial interfaces with the same wavelength and amplitude in the symmetry plane are employed. The numerical results confirm the experimental finding that the growth rate of perturbations in the symmetry plane at the linear stage in the 3D- case is much smaller than that in the 2D and 3D+ cases. The difference among them can be ascribed to the different pressure and vorticity distributions associated with the principal curvatures of the initial interface. On the one hand, the high-pressure zones in the vicinity of the deformed interface are significantly different for three cases especially in the very beginning. The shock convergence and divergence at the interface are more severe in the 3D+ case than those in the 2D case, while the wave pattern in the 3D- case is more complex. On the other hand, the baroclinic vorticity distribution plays a leading role in the interface deformation of the 3D RMI after the passage of the planar shock. The accumulated vorticity changes the movement of the deformed interface and makes the local growth of perturbations different among three cases.

Slowness surface for bulk wave propagation in anisotropic media can be divided into concave, saddle and convex regions by the parabolic lines. When a parabolic line crosses a symmetry plane, it leaves either an inflection point or a parabolic point. Surface normal at these points is associated with cuspidal point and swallowtail point, respectively, on the wave surface and in phonon focusing patterns. By examining the degeneracies in the Stroh eigenvalue equation, we have calculated the cuspidal points in cubic crystals analytically. In this work, the parabolic point and its surface normal are discussed. The main idea is to establish a connection between the parabolic point and the extraordinary transonic state that is related to a degeneracy with a multiplicity of four in the Stroh eigenvalue equation. Such a connection yields a series of simple expressions, which determine the locations of parabolic points and the corresponding swallowtail points. The result is demonstrated using phonon focusing patterns of cubic crystals, and the method also provides a tool for general discussion of the slowness surface geometry.

We introduce a method to construct parametric surfaces interpolating given finite points and a curve as a line of curvature in 3-dimensional Euclidean space. We present an existence theorem of a $C^{0}$-Hermite interpolation of surfaces possessing the given data. We show that every parameter curve of a constructed surface is a circular helix if the given curve is a circular helix. The method is validated with illustrative examples.

The adjoint curve of any Frenet curve r=r(s) is defined as the integral of binormal vector with any parameter s, of a curve r. In this paper,we analyzed the problem of finding a surfaces pencil through a given the spacelike adjoint curve as an asymptotic curve, geodesic or a line of curvature in Minkowski 3-space, respectively. We find necessary and sufficient condition for a surface pencil possessing the spacelike adjoint curve as an asymptotic curve, geodesic or line of curvature. We illustrate this method by presenting some examples.

In this paper, we study the Bisector surface, which defines the line of curvature on a surface play an importance role. Firstly, the Bisector surface constructed by a point and a space curve given in Euclidean 3-space. Then, it is investigated that the necessary and sufficient condition for directrix curve of this surface to satisfy line of curvature. After this, we classify the Bisector surfaces.

3D lattice Boltzmann simulation of droplet evaporation on patterned surfaces: Study of pinning–depinning mechanism

We shall now introduce two new measures of the curvature of a surface, called its gaussian and mean curvatures. Although these together contain the same information as the two principal curvatures, they turn out to have greater geometrical significance. The gaussian curvature, in particular, has the remarkable property, established in Chapter 10, that it is unchanged when the surface is bent without stretching, a property that is not shared by the principal curvatures. In the present chapter, we discuss some more elementary properties of the gaussian and mean curvatures, and what a knowledge of them implies about the geometry of the surface.

The classification of isoparametric hypersurfaces with four principal curvatures in the sphere interplays in a deep fashion with commutative algebra, whose abstract and comprehensive nature might obscure a differential geometer's insight into the classification problem that encompasses a wide spectrum of geometry and topology. In this paper, we make an effort to bridge the gap by walking through the important part of commutative algebra central to the classification of such hypersurfaces, such that all the essential ideal-theoretic ingredients are laid out in a way as much intuitive, motivating and geometric with rigor maintained as possible. We then explain how we developed the technical side of the entailed ideal theory, pertinent to isoparametric hypersurfaces with four principal curvatures, for the classification done in our papers~\cite{CCJ},~\cite{Ch1} and~\cite{Ch3}.