Surfaces Of Revolution Research Articles

Description and Exploration of Mean-Gauss Surfaces

In this paper we explore solving the prescribed mean curvature equation for surfaces meeting a new relation given by (H_S) = λ(K_S), where H_S and K_S are the mean and Gaussian curvatures, respectively. We prove several existence theorems for various families of surfaces and state a conjecture for surfaces of revolution. To conclude, we state a weak existence theorem, and a strong conjecture concerning possible solutions. The intention is that by using differential geometry tools which would have likely been seen at the undergraduate level, the paper and its results are more accessible. My hope is that these new theorems find applications in the classification of surfaces in the future, or at the very least serves as an interesting curiosity.

The m-polar fuzzy ELECTRE-I integrated AHP approach for selection of non-traditional machining processes

In the literature, there are a number of decision-making strategies that can be used to choose the best NTM processes. Chen (Citation2014) introduced a novel method to handle fuzzy data that includes multipolar uncertainty, referred to as the m-polar fuzzy set (mFS) approach. The mFS method, along with other multi-criteria decision-making (MCDM) techniques, is a good way to choose between options. An illustration of such a combination is the mFS elimination and choice translating reality-I (ELECTRE-I) . A criteria weight approach is also needed to increase the accuracy of the mFS ELECTRE-I method. The mFS ELECTRE-I method and the analytical hierarchy process (AHP) criteria weight calculation method are combined in the current work. The unique thing about this method is that it can be used to solve both MCDM and MCGDM problems by combining the mFS ELECTRE-I with the AHP criteria weight method. A single-dimensional weight sensitivity analysis is performed to confirm the technique’s stability for different criterion weights for the AHP method on alternative rank performance. The results of the NTM process selection are validated by previous research findings. EDM turned out to be the best way to create machine precise holes on duralumin alloy, and ECM turned out to be the best alternative to generate the surface of revolution in stainless steel. A C++-based soft solution that uses the mFS ELECTRE-I technique to analyze various MCDM and MCGDM problems has been developed. With the soft solution, you can fix problems with selecting the FMS, the robot, and so on.

NURBS-based form-finding algorithm for double-curvature cable domes

Cable domes are one of the most widely used types of cable structures due to their light weight, aesthetic view, and adaptable forms. However, feasibility and structural stability are key elements in designing new forms of cable domes. This paper uses, in a two-stage form-finding algorithm, two NURBS curves to develop a surface of translation (sweeping one curve over the other) or a surface of revolution (rotating one curve around z axis). The main aim of the algorithm is to control the curvature of the dome surface while searching for a feasible geometry, mainly in structures with unfeasible initial configurations, i.e., with zero self-stress states. Examples of these structures are the negative- and zero-Gaussian curvature cable domes of Geiger type. The algorithm starts with a global optimization technique that searches for an optimal feasible NURBS surface for the dome and followed by a local optimization technique that slightly calibrates the vertical locations of nodes to enhance the prestress equilibrium at free nodes. The NURBS surface, not only directs the form-finding process towards a smooth configuration by defining the appropriate degree of the surface, but also maintains the desired sag and rise of negative and positive curves, respectively, to the end of form-finding. The efficiency of the proposed algorithm is verified by applying it to a double-curvature cable dome from the literature. After that, a parametric study is carried out on the reference dome with multiple rise/sag ratios to investigate the effect of rise/sag on the feasibility and stability of such forms. To emphasize the significance of the algorithm in controlling the curvature of the surface, two new forms of cable domes are also developed; one is circular with zero-curvature and the other is elliptic with negative-curvature.

Realization of an Adaptation Criterion in Grid Generation Technology for Constructions Bounded by Surfaces of Revolution with Parallel Axes of Revolution1

Realization of an Adaptation Criterion in Grid Generation Technology for Constructions Bounded by Surfaces of Revolution with Parallel Axes of Revolution1

Self-similar solutions to the MCF in [formula omitted

We make an analysis of self-similar solutions for the mean curvature flow (MCF) by surfaces of revolution and ruled surfaces in R3. We prove that self-similar solutions of the MCF by non-cylindrical surfaces in R3 are trivial. Moreover, we characterize the self-similar solutions of the MCF by surfaces of revolutions under a homothetic helicoidal motion in R3 in terms of the curvature of the generating curve. Finally, we characterize the self-similar solutions for the MCF by cylindrical surfaces under a homothetic helicoidal motion in R3. Explicit families of exact solutions for the MCF by cylindrical surfaces in R3 are also given.

Percolation through voids around toroidal inclusions.

In the case of media comprised of impermeable particles, fluid flows through voids around impenetrable grains. For sufficiently low concentrations of the latter, spaces around grains join to allow transport on macroscopic scales, whereas greater impenetrable inclusion densities disrupt void networks and block macroscopic fluid flow. A critical grain concentration ρ_{c} marks the percolation transition or phase boundary separating these two regimes. With a dynamical infiltration technique in which virtual tracer particles explore void spaces, we calculate critical grain concentrations for randomly placed interpenetrating impermeable toroidal inclusions; the latter consist of surfaces of revolution with circular and square cross sections. In this manner, we study continuum percolation transitions involving nonconvex grains. As the radius of revolution increases relative to the length scale of the torus cross section, the tori develop a central hole, a topological transition accompanied by a cusp in the critical porosity fraction for percolation. With a further increase in the radius of revolution, as constituent grains become more ringlike in appearance, we find that the critical porosity fraction converges to that of high-aspect-ratio cylindrical counterparts only for randomly oriented grains.

Channel linear Weingarten surfaces in space forms

Channel linear Weingarten surfaces in space forms are investigated in a Lie sphere geometric setting, which allows for a uniform treatment of different ambient geometries. We show that any channel linear Weingarten surface in a space form is isothermic and, in particular, a surface of revolution in its ambient space form. We obtain explicit parametrisations for channel surfaces of constant Gauss curvature in space forms, and thereby for a large class of linear Weingarten surfaces up to parallel transformation.

What Is the Shape of a Cupola?

This article examines the shape of a surface obtained by a hanging flexible, inelastic material with prescribed area and boundary curve. The shape of this surface, after being turned upside down, is a model for cupolas (or domes) under the simple hypothesis of compression. Investigating the rotational examples, we provide and illustrate a novel design for a roof that has the extraordinary property that its shape, although natural, is modeled by a surface of revolution whose axis of rotation is horizontal.

Zermelo navigation problems on surfaces of revolution and geometric optimal control

In this article, the historical study from Carathéodory-Zermelo about computing the quickest nautical path is generalized to Zermelo navigation problems on surfaces of revolution, in the frame of geometric optimal control. Using the Maximum Principle, we present two methods dedicated to analyzing the geodesic flow and to compute the conjugate and cut loci. We apply these calculations to investigate case studies related to applications in hydrodynamics, space mechanics and geometry.

On singular integrals and maximal operators along surfaces of revolution on product domains

On singular integrals and maximal operators along surfaces of revolution on product domains

On the rigidity and analytical rigidity of two-connected regular surfaces of revolution for a given direction of displacement of edge points

In this paper, we investigate infinitesimal bends of two-connected surfaces of revolution on which are given conditions that allow the points of one of the boundary parallels to move only along a given constant direction. We formulated the obtained results in the form of a theorem.

Motorized Measurement of Deformation on Surface of Revolution With 2-D Laser Profiler

The 2D laser profiler potentially provides an efficient way to measure the profile of a deformed surface of revolution. However, the application is constrained by its finite measurement range, as well as the accumulated errors during device installation, workpiece placement and 3D reconstruction. In this work, an integrated measurement system is designed with high-precision rotary table (HRT) and a motorized vertical stage (MVS), so that in-situ scanning measurement is achieved by coordinated motion of the workpiece and the 2D laser profiler. An extrinsic calibration method is proposed between the 2D laser profiler and the HRT with fittings of a plane and a circle curve by the Leica absolute tracker (LAT), so that large tolerance of installation error is allowed for both MVS and HRT. Eventually, the point cloud of the workpiece with respect to the HRT is virtually relocated on the CAD model by Iterative Closest Point (ICP) algorithm with kd-tree construction, so that the amount of deformation is correctly assessed even when the deformed workpiece is arbitrarily clamped by the 3-jaw chuck. The experiment is conducted on the testbed for measuring the deformation of a wheel rim, which verifies the effectiveness of the developed system and proposed method.

Realization of the adaptation criterion in the grid generation technology for constructions bounded by the surfaces of revolution with parallel axes of revolution

Realization of the adaptation criterion in the grid generation technology for constructions bounded by the surfaces of revolution with parallel axes of revolution

On general solutions of equidistant vector fields on two-dimensional (pseudo-)Riemannian spaces

The paper is devoted to studying equidistant two-dimensional (pseudo-) Riemannian spaces. Embeddings of these spaces in three-dimensional Euclidean and Minkowski spaces as revolution or helical surfaces are given. The general solution of equidistant equations is found beyond these spaces under minimal requirements for the differentiability of the studied objects. These vector fields are associated with Killing vector fields on those spaces.

CONSTRUCTION OF GEODESIC LINES ON SURFACES OF ROTATION OBTAINED BY DISPLACEMENT OF THE MERIDIAN

Geodesic surface lines are analogous to straight lines on a plane. In addition to connecting two points of the surface by the shortest distance, they are the winding trajectories of the reinforcing threads in the strengthening of high-pressure cylinders. Just as a bundle of straight lines can be drawn from a given point on a plane in different directions, so there are geodesic lines on a surface that pass through a given point in different directions. Finding geodesic surface lines in the general case comes down to solving second-order differential equations. The purpose of the study is to investigate geodesic lines on the surface formed by the rotation of a given plane curve around a vertical axis and their transformation when this curve is shifted away from or towards the axis. For surfaces of revolution, the second-order differential equation can be reduced to the first order and even reduced to an integral based on the well-known Clerot formula. However, in this case, geodesic lines in all directions can be constructed only for a limited number of surfaces of rotation, and only limited fragments of geodesic lines can be constructed on the remaining surfaces. The article considers the construction of geodesic lines using the numerical solution of a second-order differential equation. The obtained results were visualized.

Shooting from singularity to singularity and a quasilinear <i>p</i>-Laplace-Beltrami equation with indefinite weight

For surfaces of revolution we prove the existence of infinitely many sign-changing rotationally symmetric solutions to a wide class of $ p $-Laplace-Beltrami equations with polynomial like nonlinearities. We combine the methods in [10] where the semilinear case $ (p = 2) $ with positive weight was studied with those in [5] and [6] where radial solutions to a $ p $-Laplacian Dirichlet problem in a ball was considered. Our equations lead to ordinary differential equations with two singularities and a sign changing weight function which leads to an initial value problem that may blow up in the region of definition of the equation.

Necessary conditions for existence of an additional integral in the problem on motion of a solid body with a fixed point in the flow of particles bounded by an ellipsoid revolution surface

Рассматривается задача о движении в свободном молекулярном потоке частиц твердого тела с неподвижной точкой, ограниченного поверхностью эллипсоида вращения. Получены необходимые условия существования в этом случае дополнительного аналитического первого интеграла, независимого с интегралом энергии.

On Certain Estimates for Parabolic Marcinkiewicz Integrals Related to Surfaces of Revolution on Product Spaces and Extrapolation

In this paper, appropriate Lp bounds for particular classes of parabolic Marcinkiewicz integrals along surfaces of revolution on product spaces are obtained. These bounds allow us to use Yano’s extrapolation argument to obtain the Lp boundedness of the aforesaid integral operators under weak conditions on the kernels. These conditions on the kernels are the best possible among their respective classes. In this work, several previously known results on Marcinkiewicz integrals are fundamentally improved and extended.

Analytical surfaces for architecture and engineering

Geometers have proposed more than 600 analytical surfaces for implementation. The largest number of these surfaces is used in architecture and mechanical engineering. Although digital architecture and free form architecture are now increasingly influencing the design of long-span shell structures and curved buildings, the research and application of analytical surfaces continues on an increasing scale. The purpose of the research is to study the state of affairs in the application of analytical surfaces in the construction and engineering industries and to clarify the classes of surfaces that have found application in the study of physical phenomena or in solving purely mathematical problems, but not used in other areas of human activity. Another goal is to find analytical surfaces promising for application in architecture and mechanical engineering, which are still little known to architects and engineers. It has been established that, as before, designers take new analytical surfaces to implement their creative ideas from well-studied classes of surfaces of revolution, transfer and umbrella, minimal, ruled, wavy surfaces.

Profiles of Liquid on the Surface of Revolution with Varying Cross-section under Microgravity

Profiles of Liquid on the Surface of Revolution with Varying Cross-section under Microgravity