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Consider a general reaction-diffusion problem, $u_t = \Delta u + f(x, u, u_x)$, on a revolution surface or in an $n$-dimensional ball with Dirichlet boundary conditions. In this work, we provide conditions related to the geometry of the domain and the spatial heterogeneities of the problem that ensure the existence or not of a non-constant stationary stable solution. Several applications are presented, particularly with regard to the Allen–Cahn, Fisher–KPP and sine-Gordon equations.

The article concerns the problem of covering the lateral surface of a right circular cylinder or a cone with equal balls. The surface is required to belong to their union, and the balls’ radius is minimal. The centers of the balls must lie on the covered surface. The problem is relevant for mathematics and for applications since it arises in security and communications. We develop heuristic algorithms for covering construction based on a geodesic Voronoi diagram. The construction of a covering is a non-trivial task since the line of intersection of a cylinder or a cone with a sphere is a closed curve of the fourth order. To compare the numerical results with the known ones, we unroll the surface of revolution onto a plane. Another feature is that, we use both Euclidean distance and a special non-Euclidean metric, which can describe the speed of signal propagation in a heterogeneous medium. We also perform a numerical experiment and discuss its results. Meanwhile, it is shown that with a small number of circles covering a planification of the cylindrical surface, their radius is significantly less than for a similar rectangle.

An integral geometry problem is considered on a family of n$n$‐dimensional surfaces of revolution whose vertices lie on a hyperplane and directions of symmetry axes are fixed and orthogonal to this plane, in Rn+1$\mathbb {R} ^{n+1}$. More precisely, the reconstruction of a function f(x,y)$f(\mathbf {x,}y)$, x∈Rn$\mathbf {x}\in \mathbb {R} ^{n}$, y∈R$y\in \mathbb {R}$, from the integrals of the form f(x,y)dx$f(\mathbf {x,}y) d\mathbf {x}$ extended over a chosen side of all surfaces of revolution of a given family is investigated. Unlike the usual Radon transform, the integrals considered here are not taken with respect to the surface area element. A Fourier slice identity and a backprojection‐type inversion formula are obtained with a method based on the Fourier and Hankel transforms. The reconstruction procedure and some analytical and numerical implementations of the obtained inversion formulas in the cases of n=1$n=1$ and n=2$n=2$ are provided. [ABSTRACT FROM AUTHOR]

Kondo-Tanaka proved that if a rotationally symmetric plane Mm is von Mangoldt or Cartan-Hadamard outside a compact set and has finite total curvature, then it has a sector with no pair of cut points. We show that the condition of finite total curvature can be removed. [ABSTRACT FROM AUTHOR]

Purpose: In Islamic architecture, using arches to build dome-shaped buildings has been very common. So, the research on building the domes of shrines and mosques is undoubtedly directed at studying the arches of those buildings. In this article, we will investigate and geometrically model the domes from the perspective of differential geometry and as a rotating surface. We try to present the scientific connection between the art of architecture and differential geometry in a way that interests experts in both architectural and mathematical trends. Methodology: In architecture, the dome is the product of a productive cycle around the vertical axis. This interpretation is equivalent to the definition of the rotating procedure (generating curve) in the subject of differential geometry. Special methods can obtain the generator curve. At first, according to the drawing method, we parametrize half of the arch in the Euclidean xoz plane according to the length of the dome opening and then rotate the resulting curve (or the generating curve) around the vertical z-axis. The method of conducting this research is quantitative and includes calculations related to the types of domes, and its type can be considered descriptive research. Findings: We found a significant link between the mathematics that governs domed buildings and the productive arch. Originality/Value: Considering the multitude of types of arches in architecture, in this article, after stating some necessary definitions of differential geometry, in addition to presenting the method of drawing each arch, we will only bring the calculations related to the types of arches with legs, horned goats, five-o-seven and shamrocks. Ultimately, we will implement our calculations on the dome of Juma Mosque in Ardabil. [ABSTRACT FROM AUTHOR]

We present a straightforward approach for reconstructing 3D celadon models from a single 2D image. The celadon is a historical example of the surface of revolution. Our approach uses a surface of revolution technique to generate the basic shape of the celadon and then applies texture mapping to create a realistic appearance. The process involves detecting the contour and corners of the celadon image, determining an axis of revolution, generating a profile curve, and finally constructing a 3D celadon model. Additionally, we create models as triangular meshes at multiple resolutions, employing a B-spline curve as the profile curve. It enhances the adaptability of the models for various purposes. We render various scenes using a path tracer to assess the suitability of the generated 3D celadon models and generate a VR celadon museum with the models. Overall, our approach offers a simple and efficient solution for reconstructing a 3D celadon model, generating VR content, and demonstrating extensive applicability across numerous disciplines. [ABSTRACT FROM AUTHOR]

A K α -translator is a surface in Euclidean space R 3 that moves by translations in a spatial direction under the K α -flow, where K is the Gauss curvature and α is a constant. We classify all K α -translators that are rotationally symmetric. In particular, we prove that for each α there is a K α -translator intersecting orthogonally the rotation axis. We also describe all K α -translators invariant by a uniparametric group of helicoidal motions and the translators obtained by separation of variables. [ABSTRACT FROM AUTHOR]

The work is devoted to the study of the Bianchi transform for surfac­es of revolution of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Minding top, the Minding coil, the pseudosphere (Beltrami surface). The study of surfaces of constant negative Gaussian curvature (pseudospherical surfaces) is of great importance for the interpretation of Lobachevsky planimetry. The connection of the geometric characteristics of pseudospherical surfaces with the theory of networks, with the theory of solitons, with nonlinear differential equations and sin-Gordon equations is established. The sin-Gordon equation plays an important role in modern physics. Bianchi transformations make it possible to obtain new pseudospherical surfaces from a given pseudospherical surface. The Bianchi transform for the Minding top is constructed. Using a mathematical package, Minding's top and its Bianchi transform are constructed.

We present a straightforward approach for reconstructing 3D celadon models from a single 2D image. The celadon is a historical example of the surface of revolution. Our approach uses a surface of revolution technique to generate the basic shape of the celadon and then applies texture mapping to create a realistic appearance. The process involves detecting the contour and corners of the celadon image, determining an axis of revolution, generating a profile curve, and finally constructing a 3D celadon model. Additionally, we create models as triangular meshes at multiple resolutions, employing a B-spline curve as the profile curve. It enhances the adaptability of the models for various purposes. We render various scenes using a path tracer to assess the suitability of the generated 3D celadon models and generate a VR celadon museum with the models. Overall, our approach offers a simple and efficient solution for reconstructing a 3D celadon model, generating VR content, and demonstrating extensive applicability across numerous disciplines.

In this paper we introduce the surface of revolution discrete element method (SR-DEM) for simulating systems of axi-symmetric, non-spherical particles with a closed surface of revolution. Due to the cylindrical symmetry of a surface of revolution, the geometry of any cross-section about the axis of rotation remains constant. Exploiting this geometric feature, we propose a node-to-cross-section contact algorithm to efficiently detect contact between particles with a surface of revolution. Within the SR-DEM, the contact algorithm operates in a master–handler fashion: the master particle is approximated by its surface nodes, while the handler particle is represented by a signed distance field of the cross-section about the axis of rotation. This hybrid formulation in both 2D and 3D space enables efficient contact calculations with a relatively simple implementation with a low computational cost. The SR-DEM is validated with various test cases, including particle–particle collisions, particle–wall collisions, a granular packing in a cylindrical container, and the motion of tablets in a rotating drum. Finally, we propose a straightforward approach to determine the optimal surface resolution of a non-spherical particle by increasing the number of surface nodes until the bulk properties characterizing the system converge. [Display omitted] • Node-to-cross-section algorithm to handle contacts between axi-symmetric particles. • Very efficient contact detection if the cross-section of the particles consists of few curves. • The algorithm is validated with various test-cases. • A strategy is presented to find the optimal resolution of a node-based particle surface. [ABSTRACT FROM AUTHOR]

Liquid storage structures represent an important component of modern infrastructure. They can take variety of shapes of which the conical shape is one of the most common configurations. Conical tanks are preferred by both architects and structural engineers because of their appealing look and structural efficiency in addition to their large capacities with relatively small footprint area. The state of stresses in these tanks is rather complicated and needs powerful computational tools. However, in the preliminary design phase, it is important to have a simplified analysis method for selection of economic design parameters including tank height, inclination angle, and footprint radius needed to achieve the desired tank capacity. It is also of importance that the structural engineer has an insight and understanding of the effect of these various parameters on the resulting internal forces acting on the tank. This paper presents a simplified analysis of conical tanks under hydrostatic loading based on the application of the membrane theory. The equations governing the behavior of these structures are first derived. Then, they are applied on several vessels of practical dimensions and the resulting of stresses are presented to give a deeper understanding of the resulting internal actions. Moreover, a simple guide to achieve efficient structural preliminary design parameters for a wide range of tank capacities is introduced.

In this paper we consider a three dimensional Kropina space and obtain a partial differential equation that characterizes minimal surfaces with the induced metric. Using this characterization equation we study various immersions of minimal surfaces. In particular, we obtain the partial differential equation that characterizes the minimal translation surfaces and show that the plane is the only such surface. [ABSTRACT FROM AUTHOR]

Finding out the relative equilibria and analyzing their stability is of great significance for revealing the intrinsic characteristics of mechanical system and developing effective controllers to improve system performance. In this paper, we study the symmetry and relative equilibria of a bicycle system moving on a surface of revolution. We notice that the symmetry group existing in the bicycle configuration description is a three-dimensional Abelian Lie group, and the rolling condition of the two wheels produces four time-invariant first-order linear constraints on the bicycle system. Therefore, the bicycle dynamics can be classified as a general Voronets system in which the Lagrangian and constraint distribution remain unchanged under the action of the symmetry group. Applying the Voronets equations to bicycle dynamics modeling, we obtain a seven-dimensional reduced dynamic system on the reduced constraint space. Theoretical analysis for the reduced dynamic system shows that it has the properties of time-reversal and lateral symmetries. In addition, two types of relative equilibrium points, the static equilibria and the dynamic equilibria, exist. Further theoretical analysis shows that the two kinds of relative equilibria both form one-parameter solution families, and the Jacobian matrix at an equilibrium point has some specific properties that support the relevant stability analysis. The necessary condition responsible for a stable static equilibrium point is that all the eigenvalues of the Jacobian matrix at the equilibrium point must lie on the imaginary axis of the complex plane. Due to the existence of zero eigenvalues of the Jacobian matrix, the stability of the dynamic equilibria is studied by limiting the reduced dynamic system to an invariant manifold based on the conservation of system energy. We prove in a strict mathematical sense that the dynamic equilibria may be Lyapunov stable, but cannot be asymptotically stable. Finally, symbolic computation combined with Whipple bicycle benchmark parameters was used for numerical simulations. We then use our numerical simulation to study how the parameter of the surface of revolution affects the relative equilibrium solution and its stability. [ABSTRACT FROM AUTHOR]

Aims of research. The use, design and analysis of architectural and building structures in the form of smooth and composite surfaces have become relevant and in demand lately, which determined the purpose of this article - to analyze the use of analytical surfaces given vector, parametric or explicit equations in real structures. Methods. The relationship between studies on the geometry of surfaces of revolution and transport and the creation of new forms of thin-walled structures and buildings is determined. An example of a real structure is given on each surface. The article does not consider composite, multifaceted, fractal surfaces, as well as surfaces that are not defined analytically. Results. It turned out that only a small number of considered surfaces of these two classes have found application in the world. At the end of the article, a bibliography is presented, which sets out the mathematical side of the design of analytical surfaces, their computer modeling, more detailed information about real structures in the form of the surfaces under consideration.

A ruled surface of revolution with moving axes and angles is a rational tensor product surface generated from a line and a rational space curve by rotating the line (the directrix) around vectors and angles generated by the rational space curve (the director). Only right circular cylinders and right circular cones are ruled surfaces that are also surfaces of revolution, but we show that a rich collection of other ruled surfaces such as hyperboloids of one sheet, 2-fold Whitney umbrellas, and a wide variety other interesting ruled shapes are ruled surfaces of revolution with moving axes and angles. We present a fast way to compute the implicit equation of a ruled surface of revolution with moving axes and angles from two linearly independent vectors that are perpendicular to the directrix of the surface. We also provide an algorithm for determining whether or not a given rational ruled surface is a ruled surface of revolution with moving axes and angles. [ABSTRACT FROM AUTHOR]

In this note we address the question of existence of non-constant stable stationary solution to the heat equation on surfaces of revolution subject to nonlinear boundary flux involving a positive parameter. Our result is independent of the surface geometry and, in addition, we provide the asymptotic profile of the solutions and some examples where the result applies.

Design procedure of pressure vessels is very important due to their vast applications in many industries. This procedure is mainly based on determining the stress and strain distribution, which is resulted from the internal pressure. In this paper a thin-walled pressure vessel of circular-arc cross-section is analytically studied. The vessel is a surface of revolution generated by rotating a circular arc about an axis that neither intersects the arc nor necessarily passes through the arc center. Both convex and concave vessels with open- and closed-end conditions are considered. The equilibrium equations for a proper element of the vessel surface are derived and solved analytically. Assuming small deformation and elastic behavior for the vessel, the integral constant is determined based on the end boundary conditions of the vessel. Since this type of pressure vessel was not studied in the previous literature, the results of present model are compared with similar ABAQUS Finite Element (FE) simulation. A very close agreement was observed. This evidently implies the validity of the presented model. [ABSTRACT FROM AUTHOR]

The motive of this article is to study globally defined slope metrics on surface of revolution. An introduction to first approximate slope metric and its geodesic behaviour on the surface of revolution is also discussed. [ABSTRACT FROM AUTHOR]

For Legendre curves, we consider surfaces of revolution of frontals. The surface of revolution of a frontal can be considered as a framed base surface. We give the curvatures and basic invariants for surfaces of revolution by using the curvatures of Legendre curves. Moreover, we give properties of surfaces of revolution with singularities and cones. [ABSTRACT FROM AUTHOR]

The geometry on a slope of a mountain is the geometry of a Finsler metric, called here the slope metric. We study the existence of globally defined slope metrics on surfaces of revolution as well as the geodesic's behavior. A comparison between Finslerian and Riemannian areas of a bounded region is also studied. [ABSTRACT FROM AUTHOR]

This paper discusses the geometry of a surface endowed with a slope metric. We obtain necessary and sucient conditions for any surface of revolution to admit a strongly convex slope metric. Such conditions involve certain inequalities for the derivative of the associated function on the Cartesian coordinate and the polar coordinate. In particular, we apply this result to a certain well-known surface of revolution. [ABSTRACT FROM AUTHOR]

Under consideration is the problem of reconstruction of a surface of revolution from the boundary curves of its projection. Two approaches to this problem are suggested. The first approach reduces the problem to a system of functional-differential equations. We describe in detail how to obtain this system. The second approach bases on geometrical considerations and uses a piecewiseconic approximation of the desired surface. The second method rests on the auxiliary statement on the 3D reconstruction of a straight circular cone. We give a formula for calculating the base radius of the cone. In the general case, the surface of revolution is approximated by the surface of rotation of some polygonal curve. [ABSTRACT FROM AUTHOR]

The analysis of energy balance equation for viscous laminar flow of fluid or gas in the cylindrical channel in the area (zone) of warm up bounded along the longitudinal coordinate is made. It was found that at laminar flow of fluid or gas in a round pipe, in each warm up area bounded along the longitudinal coordinate there are the areas of direct and reverse flows separated by a plane that is a locus of points where temperature is maximal for each fixed value of radial coordinate r. The received results also explain occurrence of returnable currents in plasma of the high-frequency induction discharge of atmospheric pressure. [ABSTRACT FROM AUTHOR]

We consider the equation $\Delta u+f(u)=0$ on a surface of revolution with Dirichlet boundary conditions. We obtain conditions on $f$, the geometry of the surface and the maximum value of a positive solution in order to ensure its stability or instability. Applications are given for our main results.

In the problem on rolling of a heavy round homogeneous disk on a surface of revolution with a negative Gaussian curvature, the classical nonholonomic model is used in which, at each moment, the instantaneous velocity of the current drive point of the disk touching the support is zero. Stationary motions of the disk are found. We note that, within the nonholonomic model the tangential component of the reacton for a stationary motion can be larger than the pressure force. This means that such motion in practice cannot be implemented or observed if we assume that the force that provides the no-slip condition is the dry friction force with a coefficient between zero and unity. For stationary motions of the disk the conditions of the stability in the first approximation are obtained. The results of the numerical simulation of the rolling motion of the disk without slip while the mechanical energy dissipation occurs, are presented. The purpose of these studies was to verify the adequacy of the assumed nonholonomic model of coin movements observed in practice in the entertaining coinboxes of the plastic funnel type. [ABSTRACT FROM AUTHOR]

Highlights • Real-time geometric fitting for cross sections of surface of revolution (SOR). • Real-time pose and structure recovery for SOR based on geometric fitting. • SOR shape reconstruction based on ellipse fitting. • Real-time 3D tracking of SOR with the real size of cross sections. • Geometric fitting for imaged parallel circles. Abstract This paper presents a novel ellipse fitting method to simultaneously estimate the Euclidean pose and structure of a surface of revolution (SOR) by minimizing the geometric reprojection error of the visible cross sections in image space. This geometric error function and its Jacobian matrix are explicitly derived to enable Levenberg-Marquardt (LM) optimization. With the obtained pose and structure, the Euclidean shape of a SOR can be reconstructed by generating the ellipse tangency to the apparent contour of the SOR. Given the real size of several visible cross sections, this approach can be extended to perform a real-time 3D tracking of the SOR. Additionally, this technique can be also generalized to fitting for imaged parallel circles. Sufficient experiments validate the accuracy and the real-time performance of the proposed method. [ABSTRACT FROM AUTHOR]

We study the properties of the image of a rational surface of revolution under a nonsingular affine mapping. We prove that this image has a notable property, namely that all the affine normal lines, a concept that appears in the context of affine differential geometry, created by Blaschke in the first decades of the 20th century, intersect a fixed line. Given a rational surface with this property, which can be algorithmically checked, we provide an algorithmic method to find a surface of revolution, if it exists, whose image under an affine mapping is the given surface; the algorithm also finds the affine transformation mapping one surface onto the other. Finally, we also prove that the only rational affine surfaces of rotation, a generalization of surfaces of revolution that arises in the context of affine differential geometry, and which includes surfaces of revolution as a subtype, affinely transforming into a surface of revolution are the surfaces of revolution, and that in that case the affine mapping must be a similarity.

In the present work, the notion of generalized Cheng–Yau 1-type Gauss map is proposed, which is similar to the idea of generalized 1-type Gauss maps. Based on this concept, the surfaces of revolution and the canal surfaces in the Euclidean three-space are classified. First of all, we show that the Gauss map of any surfaces of revolution with a unit speed profile curve is of generalized Cheng–Yau 1-type. At the same time, an oriented canal surface has a generalized Cheng–Yau 1-type Gauss map if, and only if, it is an open part of a surface of revolution or a torus.

Thin Wall Structures (TWS) have applications in many fields of physics and engineering. Laser-Based Powder Bed Fusion (LB-PBF) is a free-form fabrication that can easily produce the TWS in a single production step. However, due to the thin nature of this component and the interaction of thermal forces on the lateral surface, the chance of defects such as cracks and distortion is high. Therefore, this research aims to investigate the effect of power, inclination angle and the number of laser passes upon dimensional (thickness) deviation, distortion and porosity of TWS made by LB-PBF. Investigating the mentioned parameters is useful to determine the capability of LB-PBF to produce Al–Si–10Mg​ TWS as well as manufacturability of these structures. To identify the effect of inclination angle, number of laser passes and laser power on dimensional deviation, distortion and porosity a full factorial Design of Experiments (DOE) has been selected. To discuss the results statistical analyses and simulation of LB-PBF are implemented. Simulations have been carried out using computational fluid dynamic software (Flow-3D V12) and the depth and width of the meltpool are predicted. The obtained dimensions of the melt-track are then generalised based on the thickness of the TWS to confirm the accuracy of the simulation. The related rheological features of LB-PBF in the production of TWS are characterised and discussed from the results of the simulations. In this paper, the first experimentally validated simulation and experimentation of thin-walled surfaces are carried out with different inclination angles and the number of laser pass to identify the effect of these control factors, on dimensional deviation, distortion and porosity for the TWS. The meltpool phenomena for the thin wall structure are identified by fundamental rheological aspects and thermophysical properties of the material. Manufacturability and porosity are quantified for these conditions of interest, thereby providing fundamental phenomenological insight combined with practical design data for the application of TWS in LB-PBF. Results of this research show that the inclination angle and the number of laser passes in LB-PBF strongly drive the meltpool features, wall thickness, distortion and porosity. In LB-PBF of TWS, by changing the inclination angle the number of laser passes, the porosity, distortion and dimensional accuracy can be controlled. [Display omitted] • The effect of LB-PBF, and the number of tool passes upon the TWS thickness and distortion have been investigated. • The interaction of overhang geometry and dynamics shear forces for inclined samples have been studied. • The effect of the control factors on open and closed porosity has been investigated. • Single-track and multi-track melt pool geometry sizes have been simulated and compared with experimental results. [ABSTRACT FROM AUTHOR]

The problems concerning the use of the singular integral representations of the electromagnetic field for the construction of self-sufficient mathematical models and re-radiating structures. The foundations of a correct analysis of the electrodynamic antenna elements and chiral metamaterials. The mechanism accounting interelemental interaction, which is realized through the process of generalized Gauss-Seidel method. The results of the numerical solution of internal and external problems for some antenna models reemitting elements and metastructures.

In this article we study the existence of non-constant stable stationary solutions to the the diffusion equation $u_t=\hbox{div}(a \nabla u)+f(u)$ on a surface of revolution whose border is supplied with zero Neumann boundary condition. Sufficient conditions on the geometry of the surface and on the diffusivity function $a$ are given for the existence of a function f such the problem possesses such solutions.

Consider a pair (S, V ), where S is a two-dimensional surface of revolution without equators, i.e. cylinder equipped Riemannian metric of revolution, V is a central potential on S such that it keeps constant when the group of rotation acts. Also consider central potentials acting on the surfaces equipped Pseudoriemannian metric of revolution. Lets select Bertrand pairs in the set of all considered pairs | the potential has to be locking, i.e. under the influence of it all bounded orbits must be closed. Such dynamical systems are Hamiltonian ones possessed four-dimensional phase space. And one could represent Bertand pairs as five-parametric set, three parameters define the inner product of the manifold, other two define potential. It is proved that only generalized law of universal gravitation and the generalized oscillator Hook law could be locking.It is well-known that in case of closed orbit the period of moving depends on the full energy, but not depends on angular momentum (classical Gordon's theorem); in this paper we established the explicit form of this relation for Bertrand systems. In case of nonbounded orbits we calculated full time of moving, noted the infinite cases, and derived the fullness of corresponding phase flows, i.e. whether time-parameter could be continued to infinitely on the integral curves of Hamiltonian vector field of energy.We show, thatBertrand systemsin pseudoriemannian case weren't integrable by the Liouville| Arnold theorem, however the connected components of regular Liouvill folia of two first integrals energy and angular momentum stayed either torii or cylinders. We proved any folia of the foliation could be either circle or torus or cylinder or pair of cylinders. Also we constructed bifurcation diagrams of momentum map, all the diagrams is divided into areas corresponding to different types of Liouville folii. Finally it was discovered whether flows were full or not.

In this paper we present a simple method to create general 3D digital surfaces of revolution based on a 2D implicit curve of revolution (therefore not limited to a circle) and a hand-drawn generatrix. Our method can handle any sequence of Euclidean 2D points, which represents a curve, as generatrix. One can choose the topology of the surface that may have 1-tunnels, 0-tunnels or no tunnels with applications in 3D printing for instance. An online tool that illustrates the method is proposed. [ABSTRACT FROM AUTHOR]