A Square Equal-Area Map Projection with Low Angular Distortion, Minimal Cusps, and Closed-Form Solutions

A novel square equal-area map projection is proposed. The projection combines closed-form forward and inverse solutions with relatively low angular distortion and minimal cusps, a combination of properties not manifested by any previously published square equal-area projection. Thus, the new projection has lower angular distortion than any previously published square equal-area projection with a closed-form solution. Utilizing a quincuncial arrangement, the new projection places the north pole at the center of the square and divides the south pole between its four corners; the projection can be seamlessly tiled. The existence of closed-form solutions makes the projection suitable for real-time visualization applications, both in cartography and in other areas, such as for the display of panoramic images.

Power mode of axisymmetric high-speed combined extrusion of bimetallic road milling picks

The article presents a simplified three-stage model of the transition from a flat to axisymmetric deformation during high-speed, combined extrusion of bimetallic road milling picks. The transition is based on the principle of equality of areas, according to which the area of a figure during a flat deformation is replaced by an equal area of a circle. In this case, it is assumed that the kinematically possible field of the slip lines remains unchanged. This approach makes it possible to reduce the number of necessary calculations and at the same time maintain a high degree of correctness of the obtained equations. The first stage shows the transition to a special case of a flat problem, in which a stepped rectangular part is extruded from the original square blank, in which one of the sides is equal to the square side of the original blank. At the second stage of the transition, a flat-step part having square sections along its entire length is extruded from the original square-section blank. Directly at the third stage, the final transition to the axisymmetric deformation of the cutter is formed, in which the area of the square is replaced by the equal area of the circle. The dependences obtained because of solving the axisymmetric problem can be recommended for calculations in the industrial implementation of the technology of high-speed combined extrusion of bimetallic milling picks.

MODIFIED RULE OF EQUAL AREA FOR APPROXIMATION OF CONDUCTING SURFACES BY WIRE GRID IN SOLVING RADIATION PROBLEMS

A modification of the equal area rule is proposed for approximation conductive surfaces by a grid of wires when solving radiation problems. It was tested in the TALGAT system using the example of a dipole on a conductive plate, with verification in the EMPro system. It is shown that the modification of this rule gives more accurate results compared to the original one.

Influences of key parameters on flow features in the curved ducts with equal area

Curved ducts are widely used in aircraft engines to improve some capability of aero-engines. Complex internal flow characteristics would be induced by the curvature in such components. In this study, the influence of parameters, including the arc angle α, the curvature radius R i, and the height H, on the local accelerating and transonic flow in the curved ducts with equal area were studied numerically and theoretically under different nozzle pressure ratios (NPRs). The range of the Re number based on the height of the duct and the velocity at the inlet was [Formula: see text] ∼ [Formula: see text]. The shear stress transport κ-ω turbulent model was proved by the test data to suitably simulate the flow field in curved ducts because it could accurately predict the flow separations under adverse pressure gradients. The uncertainty of the pressure scan value to obtain the test data was ±0.05%. Numerical results showed that the effect of α on the flow characteristics of the curved ducts is little. The maximum Ma number in the curved section reduces with the increase of R i, and that grows with the increase of H. The range of the maximum Ma number was 1.20∼1.80. The critical NPRs, which decided the special flow features, were found in the curved ducts. The critical NPR rises with the increase of R i; however, the effect of H on the critical NPR is irregular due to the flow separations located near the lower wall induced by the large adverse pressure gradient. The theoretical results based on the small perturbation theory of transonic flow in the polar coordinate system proved that the distribution of sonic line was just dependent on the inner diameter R1, the outer diameter R2, and the arc angle θmax of the curved section. The critical mass flow and the critical NPR2 are only related to R1 and R2.

The optimization process leading to the tessellation of the first geodesic dome structure, the first Planetarium of Jena

The tessellation of the first built geodesic dome structure (the first planetarium of Jena, designed by Walther Bauersfeld, built 1922–23) has been unknown until recently. While original documentation of the tessellation has been published, the concept behind it has not been uncovered. This article presents the evolution of the final tessellation based on Bauersfeld’s hand-written notes found in the Zeiss Archives in Jena. Bauersfeld contemplated various methods of subdivision and performed detailed calculations and optimality analysis on them—preceding the theoretical studies on the tessellation of geodesic domes by almost 30 years. His key findings, relevant and comparable with later studies are highlighted. The concept of the presumably final tessellation is revealed to be the equal-area triangulation of the sphere—which has to the author’s knowledge not been considered ever since for geodesic domes. The remarkably simple algorithm applied did not result in a theoretically exact solution (well known to Bauersfeld), but as shown in this article in engineering terms it got sufficiently close. Moreover, it is concluded that the resulting tessellation excels in terms of important parameters (e.g. edge length ratio, number of different edges) compared to existing practical and theoretical solutions.

Determining the area corrections affecting the map areas in GIS applications

Nowadays, there are many area-based Geographic Information Systems (GIS) applications such as real estate valuation, land tax, farming support and cost–benefit analysis. Areas used in such applications are calculated by means of two-dimensional plane geometry. However, the computed area value is not the exact area value in the terrain. In order to calculate the exact area value of a parcel, area corrections due to various factors must be taken into account. These factors are selection of projection, slope of the terrain, elevation of the terrain and scale of the map. Selection of projection and slope of terrain are available; elevation of the terrain and scale of map are not available in all GIS software. In this study, the effect of area corrections on the area value calculated from the map is examined with sample applications and the results are presented to the GIS users. According to the results, GIS users should select the equal area projection. In addition, scale of map, elevation and slope of terrain should be taken into account in the area calculation where land measurements are not possible.

Extended Equal Area Criterion Revisited: A Direct Method for Fast Transient Stability Analysis

For transient stability analysis of a multi-machine power system, the Extended Equal Area Criterion (EEAC) method applies the classic Equal Area Criterion (EAC) concept to an approximate One Machine Infinite Bus (OMIB) equivalent of the system to find the critical clearing angle. The system-critical clearing time can then be obtained by numerical integration of OMIB equations. The EEAC method was proposed in the 1980s and 1990s as a substitute for time-domain simulation for Transmission System Operators (TSOs) to provide fast, transient stability analysis with the limited computational power available those days. To ensure the secure operation of the power system, TSOs have to identify and prevent potential critical scenarios through offline analyses of a few dangerous ones. These days, due to increased uncertainties in electrical power systems, the number of these critical scenarios is increasing, substantially, calling for fast, transient stability analysis techniques once more. Among them, the EEAC is a unique approach that provides not only valuable information, but also a graphical representation of system dynamics. This paper revisits the EEAC but from a modern, functional point of view. First, the definition of the OMIB model of a multi-machine power system is redrawn in its general form. To achieve fast, transient stability analysis, EEAC relies on approximate models of the true OMIB model. These approximations are clarified, and the EAC concept is redefined with a general definition for instability, and its conditions. Based on the defined conditions and definitions, functions are developed for each EEAC building block, which are later put out together to provide a full-resolution, functional scheme. This functional scheme not only covers the previous literature on the subject, but also allows to introduce several possible new EEAC approaches and provides a detailed description of their implementation procedure. A number of approaches are applied to the French EHV network, and the approximations are examined.

On the Hofer girth of the sphere of great circles

An oriented equator of [Formula: see text] is the image of an oriented embedding [Formula: see text] such that it divides [Formula: see text] into two equal area halves. Following Chekanov, we define the Hofer distance between two oriented equators as the infimal Hofer norm of a Hamiltonian diffeomorphism taking one to another. Consider [Formula: see text] the space of oriented equators. We define the Hofer girth of an embedding [Formula: see text] as the infimum of the Hofer diameter of [Formula: see text], where [Formula: see text] is homotopic to [Formula: see text]. There is a natural embedding [Formula: see text], sending a point on the sphere to the positively oriented great circle perpendicular to it. In this paper, we provide an upper bound on the Hofer girth of [Formula: see text].