Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group

We address the problem of integrability of the sub-Riemannian mean curvature of an embedded hypersurface around isolated characteristic points. The main contribution of this paper is the introduction of a concept of a mildly degenerate characteristic point for a smooth surface of the Heisenberg group, in a neighborhood of which the sub-Riemannian mean curvature is integrable (with respect to the perimeter measure induced by the Euclidean structure). As a consequence, we partially answer to a question posed by Danielli, Garofalo and Nhieu in [D. Danielli, N. Garofalo and D. M. Nhieu, Integrability of the sub-Riemannian mean curvature of surfaces in the Heisenberg group, Proc. Amer. Math. Soc. 140 2012, 3, 811–821], proving that the mean curvature of a real-analytic surface with discrete characteristic set is locally integrable.

Theta Surfaces

A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that any analytic surface that is the Minkowski sum of two space curves in two different ways is a theta surface. The four space curves that generate such a double translation structure are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.

A Combinatorial Approach for Constructing Lattice Structures

Lattice structures exhibit unique properties including a large surface area and a highly distributed load-path. This makes them very effective in engineering applications where weight reduction, thermal dissipation, and energy absorption are critical. Furthermore, with the advent of additive manufacturing (AM), lattice structures are now easier to fabricate. However, due to inherent surface complexity, their geometric construction can pose significant challenges. A classic strategy for constructing lattice structures exploits analytic surface–surface intersection; this, however, lacks robustness and scalability. An alternate strategy is voxel mesh-based isosurface extraction. While this is robust and scalable, the surface quality is mesh-dependent, and the triangulation will require significant postdecimation. A third strategy relies on explicit geometric stitching where tessellated open cylinders are stitched together through a series of geometric operations. This was demonstrated to be efficient and scalable, requiring no postprocessing. However, it was limited to lattice structures with uniform beam radii. Furthermore, existing algorithms rely on explicit convex-hull construction which is known to be numerically unstable. In this paper, a combinatorial stitching strategy is proposed where tessellated open cylinders of arbitrary radii are stitched together using topological operations. The convex hull construction is handled through a simple and robust projection method, avoiding expensive exact-arithmetic calculations and improving the computational efficiency. This is demonstrated through several examples involving millions of triangles. On a typical eight-core desktop, the proposed algorithm can construct approximately up to a million cylinders per second.

Modeling of Long-Range Reverberation Signal for Rough Bottom Consisting of Polygon Facets

To acquire a stable reverberation signal from an irregular ocean bottom, we derive the analytic surface integral of a scattered signal using Stokes’ theorem while approximating the bottom using a combination of polygon facets. In this approach, the delay difference in the elemental scattering area is considered, while the representative delay is used for the elemental scattering area in the standard reverberation model. Two different reverberation models are applied to a randomly generated rough bottom, which is composed of triangular facets. Their results are compared, and the scheme using analytic integration shows a converged reverberation signal, even with a large elemental scattering area, at the cost of an additional computational burden caused by a higher order approximation in the surface integral of the scattered signals.

The Task And the Construction of the Quadric

Non-degenerate surface of second order (PVP, quadric) is defined, if set to nine points, no three of which are collinear and no six of which lie in the same plane. The points in pairs can be imaginary. Analytic surface is defined, if recorded its the equation of three variables with numerical coefficients of F(x, y, z) = 0. Graphically, the surface is defined, if defined by its basic parameters — the lengths of its three mutually perpendicular main axes: large, medium and small. Depending on the position of the nine data points, one or two of the three axes can be imaginary. Analytical surface definition through writing her equations on the coordinates of the nine points is possible without any difficulties. But here we are interested in graphic design. This paper examines the question of determining the parameters of PVP — determination of the position of the surface in space and definition of its main axes upon prescribed points or other surface elements. Quadric has three main axes — it is three parameters. The center of the quadric may be shifted from the origin along each of three axes — it is three parameters. Quadric can be inclined to coordinate axes in each of the three coordinate planes. Total, quadric is determined by nine parameters. Nine points determine a quadric in form and position in space. Nine points is three trio points that define the three planes, inclusive these points. The article reviews the task of a quadric with nine points, of a conical and four points, three profile of a quadric. Solution of the definition of quadric in the case where five of the nine points lie in the same plane is proposed.

BIMEROMORPHIC GEOMETRY OF RIGID ANALYTIC SURFACES

We show that any proper rigid analytic surface admits a relatively minimal regular model. Further, we give a criterion for relative minimality of proper regular rigid analytic surfaces.