Approximation of a Great Circle by using a Circular Arc on a Mercator Chart

2017 ◽  
Vol 71 (2) ◽  
pp. 497-508
Author(s):  
Miljenko Lapaine ◽  
Tomislav Jogun
Keyword(s):  
Stereographic Projection ◽  
Great Circle ◽  
Actual Length ◽  
Circular Arc ◽  
Projection Plane ◽  
Mercator Projection

This paper describes George Biddell Airy's almost completely unknown method of approximating an orthodromic arc (great circle arc) using a circular arc in the normal aspect Mercator projection of a sphere. In addition, it is demonstrated that the centre of the circle can be defined in at least two different ways, which yields slightly different results. Airy's approach is built upon in this paper. The method of computing coordinates of Airy's circle arc centre is described. The formulae derived in the paper can be used to calculate the length of Airy's approximation of the orthodromic arc connecting two points on the sphere and on the Mercator chart. Moreover, the actual length of the orthodromic arc on the sphere and on the Mercator chart can be computed using the formulae derived in this paper. The purpose of the paper is not to suggest an application of Airy's method in navigation, but to analyse Airy's proposal and to show that a great circle arc on a Mercator chart is close to a circular arc for distances which are not too great. This property can be useful in education, having in mind that the stereographic projection is the only one that maps any circle on a sphere onto a circle in the projection plane.

A new stereographic protractor

1913 ◽  
Vol 17 (78) ◽  
pp. 46-50
Author(s):  
G. F. Herbert Smith
Keyword(s):  
Stereographic Projection ◽  
Great Circle

In the stereographic projection (fig. 1) ABA'B" is the primitive circle, having the same radius a as the corresponding sphere ; ATA'T" is the circle representing a great circle on the sphere inclined at an angle θ to the plane of the projection ; O and C are the centres of the primitive circle and the circle ATA'T' respectively. CO are joined and projected both ways to meet the circles as shown.

scholarly journals Calculation of Charts on an Oblique Gnomonic Projection by Electronic Computer

1960 ◽  
Vol 13 (3) ◽  
pp. 345-347
Author(s):  
P. B. Sarson
Keyword(s):  
Electronic Computer ◽  
Great Accuracy ◽  
Lightning Flash ◽  
Projection Plane ◽  
Mercator Projection ◽  
Straight Lines

IN the Meteorological Office, great accuracy is not usually attainable in determining or forecasting the position of significant weather features. Special projections of charts are therefore not often required; the normal conic projection with two standard parallels or (near the equator) the mercator projection is quite adequate. However, in the radiolocation of thunderstorms a chart drawn on a gnomonic projection is required. The bearings of each lightning flash within one or two thousand miles are recorded from a small number of special stations (SFERICS stations). When these bearings are plotted on a gnomonic chart by straight lines drawn from the appropriate observing station the coordinates of the source of the lightning can be quickly determined and reported through normal meteorological channels. Speed is essential and therefore the charts on which the bearings are drawn are specially designed with the SFERICS stations grouped more or less evenly about the tangential point of the projection plane of the gnomonic chart.

The Flat Earth Society

2020 ◽  
pp. 272-283
Author(s):  
Nicholas Mee
Keyword(s):  
Sixteenth Century ◽  
Stereographic Projection ◽  
Camera Obscura ◽  
Johannes Vermeer ◽  
Mercator Projection ◽  
Depth Study ◽  
Familiar World ◽  
Projection Techniques ◽  
Celestial Sphere

Chapter 25 explains the construction and use of the astrolabe with reference to Geoffrey Chaucer’s A Treatise on the Astrolabe. The astrolabe is a rotating map of the heavens constructed using a stereographic projection of the celestial sphere. The projection techniques required to create this map is reminiscent of the projections used by artists to show perspective, and it is closely related to the techniques of cartographers. The most familiar world maps are produced using the Mercator projection devised by Gerardus Mercator in the sixteenth century. Johannes Vermeer included maps in many of his paintings, most notably The Geographer and The Astronomer, and the figure in these painting might be the great microscopist Antonie van Leeuwenhoek. The architect Philip Steadman made an in-depth study of whether Vermeer employed a camera obscura when painting.

THE PECULIARITIES OF MAPS IN NATO

2020 ◽  
pp. 43-50
Author(s):  
O. Goncharenko ◽  
V. Kukol ◽  
S. Mikheli
Keyword(s):  
Stereographic Projection ◽  
Basic Data ◽  
Topographic Maps ◽  
Military Operations ◽  
Engineering Structures ◽  
Digital Maps ◽  
Mercator Projection ◽  
The Creation

The article describes basic data about the features of creating and operating of NATO maps. The article provides informationabout scale standards for NATO topographic maps. The structure of topographic maps, the order of their creation, mainpurposes, tasks, requirements according to NATO standards are considered. Topographic maps at scales of 1:25 000, 1:50 000and 1: 100 000 are created by NATO countries in accordance with national requirements while maintaining their traditionaltransition to the creation of topographic maps, but adhering to a single NATO standard for mandatory mapping of WGS -84 andUTM grids, the printing of explanation symbols and abbreviations in English and application of geographical names in Latin. Atpresent, there is a coherent NATO geopolitics for the creation of topographic special maps (including digital maps), the basicprinciple of which is that each NATO member is responsible for providing the necessary cartographic materials to its troops andNATO forces on its territory and to the globe. for planning and conducting military operations. A 1: 250,000 scale map is used tostudy and evaluate in detail individual, relatively small but important areas, when crossing water obstacles, during hostilities inlarge settlements, as well as when designing and constructing large engineering structures. Projections of topographic maps ofscale 1: 250 000 are considered, specifics of delineation and designations adopted for the map, features of the content of th etopographic map according to NATO standards. The maps are created in the Universal Transversal Mercator Projection (UTM),the Universal Polar Stereographic Projection (UPS) and the Lambert Conformal Conic Projection. The article presents a system ofgraphic symbols and symbols of NATO.

scholarly journals Radical Axes in Spherical Geometry

1884 ◽  
Vol 3 ◽  
pp. 59-61
Author(s):  
R. E. Allardice
Keyword(s):  
Spherical Geometry ◽  
Great Circle ◽  
Circular Arc ◽  
Circular Arcs ◽  
Original Figure ◽  
Point Of Intersection

It is evident that by an extension of the method of deriving from any triangle its polar triangle, it is possible to derive from any figure whatever another figure, the properties of which may be deduced at once from those of the first. This may be done either by imagining a point to move along the original figure and considering the envelope of the great circle of which the moving point is the pole; or by imagining a great circle to envelope the figure and considering the locus of its pole. In both cases two figures will be obtained; but these will be antipodal, and will therefore have like properties. Since the point of intersection of two great circular arcs is the pole of the great circular arc which joins the poles of these two arcs, the two methods of derivation mentioned above will lead to the same derived figure. These methods of transformation of figures are evidently closely analogous to that of reciprocal polars.

Charts for Great Circle Sailing

1885 ◽  
Vol 20 (501supp) ◽  
pp. 7991-7991
Author(s):  
Richard A. Proctor
Keyword(s):  
Great Circle

From Littoral to Ozone: On Mike Pearson’s Contributions to Indian Ocean History

2019 ◽  
Vol 2 (1) ◽  
pp. 12-24
Author(s):  
Edward A. Alpers
Keyword(s):  
New York ◽  
Indian Ocean ◽  
Great Circle ◽  
Historical Scholarship ◽  
The Indian Ocean ◽  
Seminal Article

In this article I examine two of Michael Pearson’s most important contributions to our understanding of Indian Ocean history: the concept of the littoral, which he first articulated in his seminal article on “Littoral society: the case for the coast” in The Great Circle 7, no. 1 (1985): 1-8, and his comment in The Indian Ocean (London and New York: Routledge, 2003, p. 9) that “I want it to have a whiff of ozone.” Accordingly, I review Pearson’s publications to see how he has written about these two notions and how they have influenced historical scholarship about the Indian Ocean.

STUDY OF INACCURACY IN THE OPERATION OF THE MAIN CONTROLLER OF STB MACHINE

2019 ◽  
Keyword(s):  
Normal Distribution ◽  
Stable Process ◽  
Random Errors ◽  
Actual Length ◽  
Experimental Data Processing ◽  
Set Up ◽  
Main Regulator ◽  
Transmission Gear ◽  
Selection Of

At production of fabrics, including fabrics for agricultural purpose, an important role is played by the cor-rect adjustment of operation of machine main regulator. The quality of setup of machine main controller is determined by the proper selection of rotation angle of warp beam weaving per one filling thread. In the pro-cess of using the regulator as a result of mistakes in adjustment, wear of transmission gear and backlashes in connections of details there are random changes in threads length. The purpose of the article is the research of property of random errors of basis giving by STB machine regulator. Mistakes can be both negative, and positive. In case of emergence only negative or only positive mistakes operation of the machine becomes im-possible as there will be a consecutive accumulation of mistakes. As a result of experimental data processing for stable process of weaving and the invariable diameter of basis threads winding of threads it is revealed that the random error of giving is set up as linear function of the accidental length having normal distribution. Measurements of accidental deviations in giving of a basis by the main regulator allowed to construct a curve of normal distribution of its actual length for one pass of weft thread. The presented curve of distribution of random errors in giving of a basis is the displaced curve of normal distribution of the accidental sizes. Also we define the density of probability of normal distribution of basis giving errors connected with a margin er-ror operation of the main regulator knowing of which allows to plan ways of their decrease that is important for improvement of quality of the produced fabrics.

The characteristics of transforming coordinates from the geocentric system WGS-84 for the Mercator projection under low latitudes conditions

2018 ◽  
Vol 940 (10) ◽  
pp. 2-6
Author(s):  
J.A. Younes ◽  
M.G. Mustafin
Keyword(s):  
Coordinate System ◽  
Satellite System ◽  
Programming Algorithm ◽  
Low Latitude ◽  
Model Calculations ◽  
Mercator Projection ◽  
Low Latitudes ◽  
Rectangular Coordinates ◽  
Global Navigation Satellite ◽  
Coordination Methods

The issue of calculating the plane rectangular coordinates using the data obtained by the satellite observations during the creation of the geodetic networks is discussed in the article. The peculiarity of these works is in conversion of the coordinates into the Mercator projection, while the plane coordinate system on the base of Gauss-Kruger projection is used in Russia. When using the technology of global navigation satellite system, this task is relevant for any point (area) of the Earth due to a fundamentally different approach in determining the coordinates. The fact is that satellite determinations are much more precise than the ground coordination methods (triangulation and others). In addition, the conversion to the zonal coordinate system is associated with errors; the value at present can prove to be completely critical. The expediency of using the Mercator projection in the topographic and geodetic works production at low latitudes is shown numerically on the basis of model calculations. To convert the coordinates from the geocentric system with the Mercator projection, a programming algorithm which is widely used in Russia was chosen. For its application under low-latitude conditions, the modification of known formulas to be used in Saudi Arabia is implemented.

scholarly journals Ductile failure analysis of epoxy resin plates containing multiple circular arc cracks by means of the equivalent material concept

2021 ◽  
Vol 16 (1) ◽  
Author(s):  
M. Pourseifi ◽  
A. S. Rahimi
Keyword(s):  
Ductile Failure ◽  
Equivalent Material ◽  
Circular Arc ◽  
Equivalent Material Concept ◽  
Crack Tips ◽  
Fracture Tests ◽  
Polymeric Samples ◽  
Cracked Specimens ◽  
Brittle Fracture Criterion ◽  
Material Concept

AbstractDuctile failure of polymeric samples weakened by circular arc cracks is studied theoretically and experimentally in this research. Various arrangements of cracks with different arc angles are considered in the specimens such that crack tips experienced the mixed mode I/II loading conditions. Fracture tests are conducted on the multi-cracked specimens and their fracture loads are achieved. To provide the results, the equivalent material concept (EMC) is used in conjunction of dislocation method and a brittle fracture criterion such that there is no necessity for performing complex and time-consuming elastic-plastic damage analyses. Theoretical and experimental stress intensity factors are computed and compared with each other by employing the fracture curves which demonstrate the appropriate efficiency of proposed method to predict the tests results.

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