The Geometry of a Randers Rotational Surface with an Arbitrary Direction Wind

In the present paper, we study the global behaviour of geodesics of a Randers metric, defined on Finsler surfaces of revolution, obtained as the solution of the Zermelo’s navigation problem. Our wind is not necessarily a Killing field. We apply our findings to the case of the topological cylinder R×S1 and describe in detail the geodesics behaviour, the conjugate and cut loci.

Rotational symmetry of Weingarten spheres in homogeneous three-manifolds

Let M be a simply connected homogeneous three-manifold with isometry group of dimension 4, and let Σ be any compact surface of genus zero immersed in M whose mean, extrinsic and Gauss curvatures satisfy a smooth elliptic relation {\Phi(H,K_{e},K)=0}. In this paper we prove that Σ is a sphere of revolution, provided that the unique inextendible rotational surface S in M that satisfies this equation and touches its rotation axis orthogonally has bounded second fundamental form. In particular, we prove that: (i) Any elliptic Weingarten sphere immersed in {\mathbb{H}^{2}\times\mathbb{R}} is a rotational sphere. (ii) Any sphere of constant positive extrinsic curvature immersed in M is a rotational sphere. (iii) Any immersed sphere in M that satisfies an elliptic Weingarten equation {H=\phi(H^{2}-K_{e})\geq a>0} with ϕ bounded, is a rotational sphere. As a very particular case of this last result, we recover the Abresch–Rosenberg classification of constant mean curvature spheres in M.

On the Formulation of Nonreflecting Boundary Conditions for Turbomachinery Configurations: Part I — Theory and Implementation

With unsteady flow simulations of industrial turbomachinery configurations becoming more and more affordable there is a growing need for accurate inlet and outlet boundary conditions as numerical reflections alone can lead to incorrect trends in engine efficiency, noise and aeroelastic analysis parameters. This is the first of two papers on the formulation of unsteady boundary conditions which have been implemented for both time-domain and frequency-domain solvers. Giles’ original idea for steady solvers to formulate the boundary condition in terms of characteristics generalizes to frequency-domain solvers. The boundary condition drives the value of the incoming characteristics to ideal values that are computed using the modal decomposition of linearized 2D Euler flows. The present paper explains how to generalize 2D nonreflecting boundary conditions to real 3D annular domains by applying them in certain conical rotational surfaces. For a flow with zero radial component and an annular boundary that is perpendicular to the machine axis, these surfaces are the cylindrical streamsurfaces. For more general flows and geometries, however, there is no natural choice for the rotational surfaces. In this paper, two choices are discussed: the surfaces that are generated by the boundary normals and those that are defined by the circumferentially averaged meridional velocity. The impact of the boundary condition on the stability of the harmonic-balance solver is analyzed by studying the pseudo-time evolution of certain energy integrals. For a model problem which consists of a small disturbance of an inviscid flow, the increase or decrease of this energy integral is shown to be directly related to the normal characteristic variables along the boundary. This shows that the actual boundary condition should be formulated as a control problem for the normal characteristics. Moreover, the application of the harmonic balance solver to a simple duct configuration with prescribed disturbances demonstrates that using the characteristics based on the meridional velocity may prevent the solver from converging. In contrast, the 2D theory can be formulated in a different surface without impairing the robustness of the overall approach. These findings are illustrated by a simple test case. The impact of the choice of the rotational surface for the 2D theory is studied for various duct segments and a low-pressure turbine configuration in the second paper. There it is shown that applying the 2D theory to the meridional-velocity surfaces may be advantageous in that it leads to more accurate results.

Extraction of Rotational Surfaces and Generalized Cylinders from Point-Clouds Using Section Curves

In industrial facilities, there are various types of equipment composed of surfaces that have a high degree of freedom. Rotational surfaces and generalized cylinders are often used for equipment handling liquids and gases. In this paper, we propose methods for reconstructing rotational surfaces and generalized cylinders from noisy and incomplete point-clouds captured by a terrestrial laser scanner. In our method, we convert point-clouds into wireframe models and calculate the intersection points with section planes. Then, we extract ellipses from the intersection points on each section plane and reconstruct the rotational surfaces and generalized cylinders using the extracted ellipses. We also propose a method for subdividing a rotational surface into primitive surfaces. We evaluated our method using actual point-clouds of engineering facilities and confirmed that our method could successfully reconstruct rotational surfaces and generalized cylinders.

Isometric Deformation of (m,n)-Type Helicoidal Surface in the Three Dimensional Euclidean Space

We consider a new kind of helicoidal surface for natural numbers ( m , n ) in the three-dimensional Euclidean space. We study a helicoidal surface of value ( m , n ) , which is locally isometric to a rotational surface of value ( m , n ) . In addition, we calculate the Laplace–Beltrami operator of the rotational surface of value ( 0 , 1 ) .

A Graphical Method for Great Circle Routes

A great circle route (GCR) is the shortest route on a spherical earth model. Do we have a visual diagram to handle the shortest route? In this paper, a graphical method (GM) is proposed to solve the GCR problems based on the celestial meridian diagram (CMD) in celestial navigation. Unlike developed algebraic methods, the GM is a geometric method. Appling computer software to graph, the GM does not use any equations but is as accurate as using algebraic methods. In addition, the GM, which emphasizes the rotational surface, can depict a GCR and judge its benefit.

Three-DOF Electrostatic Induction Actuator Providing Translational and Rotational Surface-Drive Motion

This paper describes the driving characteristics of a three degree-of-freedom (three-DOF) electrostatic induction actuator, which can demonstrate surface-drive characteristics with translational and rotational motions. It consists of a sheet-type slider without electrodes and a planar stator with an array of three-phase driving electrodes. The electrodes with different orientations are aligned in a regular manner to construct a four-by-four checkerboard pattern. Controlling applied voltage patterns can generate translational or rotational patterns of electrostatic fields, which drive the slider. The performance of the three-DOF actuator with regards to translational and rotational motion was investigated.