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We consider spacelike and timelike surfaces satisfying an interesting geometric property that the gradient of the mean curvature is a principal direction in a Minkowski 3-space. It is proved that these surfaces are linear Weingarten surfaces and contain all biconservative surfaces in a Minkowski 3-space. We call them generalized biconservative surfaces (or GB surfaces for short). We give complete explicit classifications of spacelike and timelike GB surfaces in Minkowski 3-space. Our results show that the non-constant-mean-curvature GB surfaces in Minkowski 3-space are locally either surfaces of revolution or null scrolls.

Dome is often used by architects for cover of large spans. Only spherical, conical, elliptical, parabolic, and hyperbolic surfaces of revolution among tens of the well-known surfaces of revolution can be taken for middle surfaces of domes. Spherical domes have the most spread in modern building due to simplicity of their form in comparison with other shells of double curvature. But researches of paraboloidal domes do not end. Some new information on strength analysis of parabolic shells, on determination of the frequencies of their natural vibrations and the examples of application of paraboloid of revolution in civil engineering in 1900=2017 are presented in this review paper. The main presented bibliography was published in XXI century.

The paper considers stationary magnetohydrodynamic flow of viscous fluid in the slot between fixed curvilinear surfaces of revolution exposed to azimuthal magnetic field. To solve the problem, the equations of boundary layer in the curvilinear coordinate system. x,θ,y, were applied. The equations of the boundary layer were solved analytically with the use of the small-parameter method. The formulas determine the field of velocity and pressure.

A general mass transport equation for diffusion-controlled whisker growth on a surface of revolution is formulated. Two limiting cases for the whisker growth of a circular cylinder-like whisker are analyzed; one surface is planner, and the other one is spherical. The growth rate is proportional to the concentration difference for the growth of the whisker on both the planar surface and the spherical surface. Using the relation between mechanical work and chemical energy, the growth rate is found to be proportional to the pressure difference. Assuming the whisker growth as the mechanism for stress relaxation of a thin film, the whisker length and the stress relaxation associated with the whisker growth are found to be exponential functions of time.

A novel recognition of ellipses and relative pose estimation algorithm is presented based on surface of revolution (SOR) in this paper for images of multiple ellipses. Firstly, the contours following method is performed on image to detect ellipses. Secondly, candidate ellipses projected by circular cross section of a SOR are obtained based on parallel constraint and vertical constraint in the image. Finally, relative pose between target and camera is calculated by these candidate ellipses. Experimental results indicate that the method perform well in recognition and estimation. The precision of recognition is higher than 97% for synthetic images corrupted by 0 ∼ 16% salt-and-pepper noise. The absolute error of pose angle is 1°,and the absolute error in axis z and other axis are less than 80 mm and 15 mm, respectively, when measure distance is less than 10 m.

We establish the bounds of Marcinkiewicz integrals associated to surfaces of revolution generated by two polynomial mappings on Triebel–Lizorkin spaces and Besov spaces when their integral kernels are given by functions $\unicode[STIX]{x1D6FA}\in H^{1}(\text{S}^{n-1})\cup L(\log ^{+}L)^{1/2}(\text{S}^{n-1})$. Our main results represent improvements as well as natural extensions of many previously known results.

We present a family of superintegrable (SI) sytems living on a riemannian surface of revolution and which exhibits one linear integral and two integrals of any integer degree larger or equal to 2 in the momenta. When this degree is 2 one recovers a metric due to Koenigs. The local structure of these systems is under control of a linear ordinary differential equation of order n which is homogeneous for even integrals and weakly inhomogeneous for odd integrals. The form of the integrals is explicitly given in the so-called simple case (see definition 2). Some globally defined examples are worked out which live either in H2 or in R2., Comment: 43 pages

Descriptive geometry, as the elementary one, studies the real world by its abstractions. But Euclid’s geometry of the real world is conjugated to pseudo-Euclidean geometry, and they make a conjugated pair. As a consequence, each real figure is conjugated with some imaginary pattern. This paper apart from some science facts demonstrates the presence of imaginary patterns in geometric constructions, where the imaginary patterns manifest themselves as singularities or as geometrically imaginary points (GIP) in “Real — Imaginary” conjugate pairs. The study is conducted, as a rule, from simple to complex, from particulars to generals. Rotation of a circle around an arbitrary axis generates, in the general case, a quartic surface. Among the quartic surfaces are a circular torus and a sphere as a special case of the torus. The torus is obtained from the circle rotation around an axis lying in the circle plane. If the axis does not intersect the generating circle, then the surface is called an open torus; when the axis intersects the generating circle, then the surface is called a closed torus; when the rotation axis passes through the center of the generating circle, then the surface is a sphere. The open torus is associated with a bagel, and the closed one — with an apple. The torus is a perfect example for the application of two well-known Guldin’s formulas. Next, the imaginary torus support is considered in this paper, at the end of which the sphere and its imaginary sup - port are considered. Imaginary patterns lead to the complex numbers, in regards to which grieved the great J. Steiner, calling them "hieroglyphs of analysis". But imaginary patterns exist apart from analysis formulas — they are the part of geometry. J.V. Poncelet was the first who understood the imaginary points in 1812, being in Russian captivity in Saratov and, what is important, without analysis formulas at all. Computational geometry often shows quantities, large numbers of real figures, because it takes into account the imaginary images too.

We introduce some new approaches for the graphical assessment of distribution of the data that supplement the existing graphical methods. Analogous to Q–Q plots and P–P plots, we introduce plots based on arc length and area of surface of revolution of the density function. Thus, our method indirectly makes use not only of density assumed but also of the derivatives thereof. We illustrate, by using several examples, that these plots help us identify the correct distribution and also rule out the incorrect possibilities. We further consider the problem of assessing the behavior of the data toward the tail and develop graphical tools to identify the closest potential probability distribution for the tail. Examples based on real data are provided.

This paper presents a novel segmentation algorithm for mechanical CAD models (represented by either mesh or point cloud) constructed from planes, cylinders, cones, spheres, tori and easily extendable to surfaces of revolution. Our proposed approach differs from existing techniques in the following aspects. First, by assuming that common mechanical models only have a limited number of dominant orientations that their primitives are either parallel or orthogonal to, we narrow down the search space for detecting the primitives to the automatically estimated major orientations of the input model. Second, we employ a dimension reduction method which transforms the problem of detecting 3D primitives into the classical 2D problems such as circle and line detection in images. Third, we generate an over-complete set of primitives and formulate the segmentation as a set cover optimization problem. We demonstrate our method's robustness to noise and show that it compares favorably with state-of-the-art solutions such as the RANSAC-based (Schnabel et al., 2007) and GlobFit (Li et al., 2011) approaches on many synthetic and real scanned examples.

In this article, the mathematical foundations of the so called Criterion of Optimization by Compensation for designing commercial bottles with a straight section along its silhouette and with lateral surfaces of revolution is presented. Such mathematical model uses as main tools, Lagrange polynomial interpolation and Newton’s Method for Nonlinear Systems being first necessary to formulate and demonstrate a theorem. It was redesigned and manufactured a bottle of a half-liter of Fanta soda of the well-known Coca Cola Company, which uses 18.86% more material that such criterion establishes. It was expected that the redesigned bottle use 4.91% more of material with respect to what is established by the Criterion of Optimization by Compensation. However, it was reported a 13% of mistake due to important limitations that must be overcome.

In this paper we continue the analysis of equivariant wave maps from 2-dimensional hyperbolic space H² into surfaces of revolution N that was initiated in [12, 13]. When the target N = H² we proved in [12] the existence and asymptotic stability of a 1-parameter family of finite energy harmonic maps indexed by how far each map wraps around the target. Here we conjecture that each of these harmonic maps is globally asymptotically stable, meaning that the evolution of any arbitrarily large finite energy perturbation of a harmonic map asymptotically resolves into the harmonic map itself plus free radiation. Since such initial data exhaust the energy space, this is the soliton resolution conjecture for this equation. The main result is a verification of this conjecture for a nonperturbative subset of the harmonic maps., National Science Foundation (U.S.) (Grant DMS-1302782), National Science Foundation (U.S.) (Grant 1045119)

The question of whether a closed Riemannian manifold has infinitely many geometrically distinct closed geodesics has a long history. Though unsolved in general, it is well understood in the case of surfaces. For surfaces of revolution diffeomorphic to the sphere, a refinement of this problem was introduced by Borzellino, Jordan-Squire, Petrics, and Sullivan. In this article, we quantify their result by counting distinct geodesics of bounded length. In addition, we reframe these results to obtain a couple of characterizations of the round two-sphere., Comment: 7 pages

In this paper the steady laminar flow of viscous incompressible ferromagnetic fluid is considered in a slot between fixed surfaces of revolution having a common axis of symmetry. The boundary layer ferromagnetic equations for axial symmetry are expressed in terms of the intrinsic curvilinear orthogonal coordinate system x, θ ,y.The method of perturbation is used to solve the boundary layer equations. As a result, the formulae defining such parameters of the flow as the velocity components vx, vy, and the pressure , were obtained.

The paper develops a study of closed geodesics on piecewise smooth constant curvature surfaces of revolution initiated by I.V. Sypchenko and D. S. Timonina. The case of constant negative curvature is considered. Closed geodesics on a surface formed by a union of two Beltrami surfaces are studied. All closed geodesics without self-intersections are found and tested for stability in a certain finite-dimensional class of perturbations. Conjugate points are found partly.

One of the key estimates in the original paper [3] does not seem to hold true, which may invalidate some results of that paper. However, there is a couple of ways to fix this mistake by adding additional hypotheses in Theorem 1 [3] .

Bu tez çalışmasında Öklid uzayındaki conchoid eğrileri ve yüzeyleri ele alınmıştır. Bu tez dört bölümden oluşmaktadır. Birinci bölüm giriş bölümüdür. İkinci bölümde bu çalışmanın ilerleyen bölümlerinde kullanılacak olan Öklid uzayındaki eğri ve yüzeylerle ilgili bazı temel tanımlar ve kavramlar ele alınmıştır. Üçüncü bölümde düzlemde ve 3-boyutlu uzayda conchoid eğrileri tanımlanmıştır. Ayrıca bu eğrilerin eğrilikleri hesaplanıp bunlarla ilgili sonuçlar verilmiştir. Elde edilen sonuçlara göre bazı örnekler verilip grafikleri çizdirilmiştir. Dördüncü bölümde 3 ve 4-boyutlu Öklid uzayında conchoidal yüzeyler çalışılmıştır. İlk olarak 3-boyutlu Öklid uzayında daha önceden verilen conchoidal yüzey tanımına bağlı olarak eğrilikleri ile ilgili sonuçlar elde edilmiştir. Ayrıca 3-boyutlu uzayda bir conchoid eğrisinin döndürülmesi ile elde edilen dönel yüzey ile ilgili sonuçlar verilip grafikleri çizdirilmiştir. Son olarak 4-boyutlu Öklid uzayındaki conchoidal yüzey tanımı verilip bu yüzeylerin düz ve minimal olmaları ile ilgili sonuçlar elde edilmiştir. Ayrıca 4-boyutlu uzayda rotasyonel yüzeyler ve meridyen yüzeylerinin conchoidal yüzey olması örnek olarak verilmiştir. In this thesis, conchoid curves and surfaces in Euclidean space are considered. This thesis consists of four chapters. The first chapter is the introduction. The second chapter contains some well-known definitions and terms about curves and surfaces in Euclidean space which will be used in other chapters. In the third chapter, conchoid curves are defined on plane and 3- dimensional Euclidean space. Furthermore, the curvatures of these curves are calculated and the results are given. According to the results, some examples are given and plot their graphics. In the fourth chapter, conchoidal surfaces are studied in 3 and 4- dimensional Euclidean space. Firstly, in the 3- dimensional Euclidean space, the results of the curvature of the conchoidal surface are obtained. Also, the results are given of the surface of revolution obtained by rotating a planar conchoid curve in 3- dimensional space and are plotted the graphics of the surface. Finally, the definition of the conchoidal surface in the 4- dimensional Euclidean space is given. Also some results are obtained that these surfaces become flat and minimal. In the last part the rotational surfaces in the 4- dimensional space and the meridian surfaces are conchoidal surfaces are given as examples.

Affine rotation surfaces, which appear in the context of affine differential geometry, are generalizations of surfaces of revolution. These affine rotation surfaces can be classified into three different families: elliptic, hyperbolic and parabolic. In this paper we investigate some properties of algebraic parabolic affine rotation surfaces, i.e. parabolic affine rotation surfaces that are algebraic, generalizing some previous results on algebraic affine rotation surfaces of elliptic type (classical surfaces of revolution) and hyperbolic type (hyperbolic surfaces of revolution). In particular, we characterize these surfaces in terms of the structure of their implicit equation, we describe the structure of the form of highest degree defining an algebraic parabolic affine rotation surface, and we prove that these surfaces can have either one, or two, or infinitely many axes of affine rotation. Additionally, we characterize the surfaces with more than one parabolic axis.

The article presents approaches to the formation of a general computational scheme for modeling (simulating) the particle motion on an axisymmetric rotating curved surface with a vertical axis of rotation. To describe the complex particle motion over a given surface, the fundamental equation of particle dynamics in a non-inertial reference frame was used, and by projecting it onto the axes of cylindrical coordinates, the Lagrange&rsquo, s differential equations of the first kind were obtained. According to the proposed algorithm in C#, an application was developed that enables graphical and numerical control of the calculation results. The program interface contains six screen forms with tabular baseline data (input) and a table of a step-by-step calculation of results (output), particle displacement, velocity, and acceleration diagrams constructed along the axes of the system of cylindrical coordinates &rho, and z, graphical presentation of the generate of the surface of revolution and the trajectory of the absolute motion of a particle over the axisymmetric rotating surface developed in polar coordinates. Examples of the calculation of the particle motion are presented. The obtained results can be used for the study and design of machines, for example, centrifugal rotary chopper machines.

This paper presents a new rigorous approach based on the method of analytical regularization developed for analysis of scalar wave diffraction by acoustically soft doubly-connected arbitrary shaped surfaces of revolution. The surface contains two apertures of equal angular size and is formed by rotation of a smooth generating curve. This approach is validated by comparison with previously known results.

In this article, we consider surfaces in the 3dimensional Euclidean space E3without parabolic points which are of finite III-type, that is, they are of finite type, in the sense of B.-Y. Chen, with respect to the third fundamental form. We present an important family of surfaces, namely, surfaces of revolution in E3. We study a special case of this family, namely, surfaces of revolution where the sum of the radii of the principal curvature R is constant.

In this paper, we investigate translating solitons for the inverse mean curvature flow (IMCF), which is a special solution deformed only for translation under the flow. The IMCF has been studied extensively not only as a type of a natural geometric flow, but also for obtaining various interesting geometric inequalities. We show that the translating solitons that are either ruled surfaces or translation surfaces are cycloid cylinders, and completely classify 2-dimensional helicoidal translating solitons and the higher dimensional rotationally symmetric translating solitons using the phase-plane analysis. The surface foliated by circles, which is called a cyclic surface, is regarded in terms of being the translating soliton for the IMCF, and then it is a surface of revolution whose revolution axis is parallel to the translating direction. In particular, we extend the result to a higher dimension, namely, the n-dimensional translating soliton foliated by spheres lying on parallel hyperplanes in $$\mathbb {R}^{n+1}$$ must be a rotationally symmetric hypersurface whose rotation axis is parallel to the translating direction.

Within the presented research rods exhibiting the maximum buckling resistance are looked for. It was assumed that they have to possess the same mass and the same length as a solid cylindrical bar treated as the reference one. Nonprismatic rods of circular hollow cross-section and of variable wall thickness were only included into considerations. It was assumed that the external and internal surfaces of revolution were defined by some assumed in advance smooth functions. Three different classes of functions were used in the research, and namely: polynomial of the second order, sine and hyperbolic cosine functions. The classical optimization problem was defined by the objective function which was the buckling force, being maximized with appropriate constraints. Design variables were three independent geometrical parameters defining uniquely the sought bar shape. All derivations were carried out analytically by means of MathematicaTM system. The presented procedure was illustrated by two examples. The obtained increase of the critical force was dependent on the slenderness of the reference bar and reached the value 40 in the case of the longer, considered reference bar.Within the presented research rods exhibiting the maximum buckling resistance are looked for. It was assumed that they have to possess the same mass and the same length as a solid cylindrical bar treated as the reference one. Nonprismatic rods of circular hollow cross-section and of variable wall thickness were only included into considerations. It was assumed that the external and internal surfaces of revolution were defined by some assumed in advance smooth functions. Three different classes of functions were used in the research, and namely: polynomial of the second order, sine and hyperbolic cosine functions. The classical optimization problem was defined by the objective function which was the buckling force, being maximized with appropriate constraints. Design variables were three independent geometrical parameters defining uniquely the sought bar shape. All derivations were carried out analytically by means of MathematicaTM system. The presented procedure was illustrated by two examples. The obtain...

En este trabajo se hace una introducci´on a las superficies de curvatura media nula, las superficies minimales. Estas se encuentran en la naturaleza en forma de pel´ıculas jabonosas optimizando el ´area seg´un el principio de m´ınima acci´on de la Mec´anica Lagrangiana. Partiendo de la justificaci´on del comportamiento de estas pel´ıculas, definiremos las superficies minimales y mostraremos algunas caracterizaciones locales de car´acter anal´ıtico usando la ecuaci´on de Euler-Lagrange o la noci´on de funci´on arm´onica. Veremos adem´as que toda superficie de ´area m´ınima con borde prefijado es minimal y que el rec´ıproco no es cierto en general. El trabajo finaliza con el estudio de los ejemplos m´as notables de superficies minimales, dibuj´andolas y calculando algunos par´ametros como su curvatura media, su curvatura de Gauss y el ´area. Para ello utilizaremos el software matem´atico Sagemath. In this dissertation we will give an introduction to null mean curvature surfaces, which are well-known as minimal surfaces. These surfaces can be found in the nature like soap films due to their ability to minimize areas following the minimum action principle of the Lagrangian Mechanics. On the basis of the behaviour of soap films, we will define minimal surfaces, showing some local and analytic characterizations using the Euler-Lagrange equation or harmonic functions. Furthermore, we will show that every area-minimizing surface with fixed bound is a minimal surface althought the converse is not true, in general. In the last chapter, we show the most relevant minimal surfaces examples, drawing them and calculating several parameters with the mathematical software Sagemath. For example, their mean curvature, their Gauss curvature and areas.

A solution of the nonlinear integrodifferential equation for the mass transport defines the shape of a dendrite, growing in a supersaturated alloy. We obtained a numerical solution of this equation in the limit of large chemical Péclet number in the case when dendrite shape is a surface of revolution. Our solution is in good agreement with an asymptotical expression for the shape of growing crystal. © 2019 Author(s).

For a free particle that non-relativistically moves on a curved surface, there are curvature-induced quantum potentials that significantly influence the surface quantum states, but the experimental results in topological insulators, whenever curved or not, indicate no evidence of such a potential, implying that there does not exist such a quantum potential for the relativistic particles, constrained on the surface or not. Within the framework of Dirac quantization scheme, we demonstrate a general result that for a Dirac fermion on a two-dimensional curved surface of revolution, no curvature-induced quantum potential is permissible., Comment: 6 pages, no figures. Typos corrected

Here we present new $L^2$ injectivity results for 2-D and 3-D Compton scattering tomography (CST) problems in translational geometries. The results are proven through the explicit inversion of a new toric section and apple Radon transform, which describe novel 2-D and 3-D acquisition geometries in CST. The geometry considered has potential applications in airport baggage screening and threat detection. We also present a generalization of our injectivity results in 3-D to Radon transforms which describe the integrals of the charge density over the surfaces of revolution of a class of $C^1$ curves., Comment: 16 pages, 2 figures

International audience; The calculation of the diffusion-controlled reaction rate for partially absorbing, non-spherical boundaries presents a formidable problem of broad relevance. In this paper we take the reference case of a spherical boundary and work out a perturbative approach to get a simple analytical formula for the first-order correction to the diffusive flux onto a non-spherical partially absorbing surface of revolution. To assess the range of validity of this formula, we derive exact and approximate expressions for the reaction rate in the case of partially absorbing prolate and oblate spheroids. We also present numerical solutions by a finite-element method that extend the validity analysis beyond spheroidal shapes. Our perturbative solution provides a handy way to quantify the effect of non-sphericity on the rate of capture in the general case of partial surface reactivity.

Highly developed production technologies facilitate unique component characteristics depending on the respective manufacturing objective. However, often several technologies are required in a sequential manner to achieve either the geometrical or the mechanical component properties. Innovative stainless steel alloys show an outstanding work hardening capability. Therefore, they are of great interest for the development of an enhanced burnishing process. The incremental cold forming technology of profile cross rolling can be applied to burnish machined preforms. This paper examines an approach for the geometrical preform design. Starting from the geometrical surface of revolution, the procedure is further developed to helical component surfaces. Both numerical and experimental results indicate that the preform design allows attaining a hardness increase in highly loaded component subsurface regions. This hardness increase goes along with low geometrical deviations and high surface integrity. Hence the results allow the conclusion that this technological development enables the achievement of several component characteristics with one fast burnishing operation. This might lead to an increased acceptance of forming technologies in industries that are usually characterized by pure machining processes. Such an industry could be plastics compounding.

A technology for structured grid generation in the volumes bounded by surfaces of revolutions is suggested. The technology is designed for modeling the processes of multi-component hydrodynamics. The technology is developed within variational approach for generation of optimal grids. It includes several types of algorithms for different types of constructions. The basic construction is the volume of revolution obtained by the rotation through 180° about the axis of a generatrix consisting of straight line segments, arcs of circles and ellipses. The generalization of the volumes of revolution (volumes formed by the surfaces of revolution with parallel axes) and constructions obtained by the deformation of the volume of revolution by the another volume of revolution or its generalization are also considered. Algorithms perform the description of the geometry of constructions, generation of initial grids, correction of grids to the boundary of constructions, deformation of grids and constructions, optimization and testing of grids.

In this work, we obtain appropriate sharp bounds for a certain class of maximal operators along surfaces of revolution with kernels in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{q}(\mathbf{S}^{n-1})$\end{document}Lq(Sn−1), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q > 1$\end{document}q>1. By using these bounds and using an extrapolation argument, we establish the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}$\end{document}Lp boundedness of the maximal operators when their kernels are in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L(\log L)^{\alpha}(\mathbf{S}^{n-1})$\end{document}L(logL)α(Sn−1) or in the block space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B^{0,\alpha-1}_{q} (\mathbf{S}^{n-1})$\end{document}Bq0,α−1(Sn−1). Our main results represent significant improvements as well as natural extensions of what was known previously.

In this paper, we study the evolution scenarios of surfaces of revolution associated with the kink-type solutions of an integrable equation, which is called the SIdV equation because of its scale-invariant property and relationship with the Korteweg-de Vries equation, where the kink-type solutions play the role of a metric. We put forward two kinds of evolution scenarios for surfaces of revolution associated with two types of kink-type metric (solution) and we study the key properties of these surfaces.

We present an algorithm for extracting the axis of revolution from the implicit equation of an algebraic surface of revolution based on three distinct computational methods: factoring the highest order form into quadrics, contracting the tensor of the highest order form, and using univariate resultants and gcds. We compare and contrast the advantages and disadvantages of each of these three techniques and we derive conditions under which each technique is most appropriate. In addition, we provide several necessary conditions for an implicit algebraic equation to represent a surface of revolution.