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

Published online by Cambridge University Press:  29 August 2017

Miljenko Lapaine  and
Tomislav Jogun
Miljenko Lapaine*
Affiliation:
(University of Zagreb, Faculty of Geodesy, Kačićeva 26, 10000 Zagreb, Croatia)
Tomislav Jogun
Affiliation:
(University of Zagreb, Faculty of Geodesy, Kačićeva 26, 10000 Zagreb, Croatia)
*
(E-mail: mlapaine@geof.hr)
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Abstract

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.

Keywords

George Biddell AiryOrthodromeMercator projection
Type
Research Article
Information
The Journal of Navigation , Volume 71 , Issue 2 , March 2018 , pp. 497 - 508
Copyright
Copyright © The Royal Institute of Navigation 2017 

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References

REFERENCES

Airy, G.B. (1826). Mathematical Tracts on Tke Lunar and Planetary Theories, The Figure of the Earth, Presession and Nutation, The Calculus of Variations, and The Undulator Theory of Optics, J. Deighton & Sons, Cambridge.Google Scholar
Airy, G.B. (1845). The Figure of the Earth. In: Encyclopaedia Metropolitana, or Universal Dictionary of Knowledge, Editors: Smedley, Edward, Rose, Hugh James, Rose, Henry John, Volume V., London.Google Scholar
Airy, G.B. (1858). On a Method of very approximately representing the Projection of a Great Circle upon Mercator's Chart, Monthly Notices of the Royal Astronomical Society, 18(5), 150–155. doi: 10.1093/mnras/18.5.150 http://mnras.oxfordjournals.org/content/18/5/150.full.pdf+html Google Scholar
Airy, G.B. (1896). Autobiography of Sir George Biddell Airy, K.C.B., M.A., Ll.D., D.C.L., F.R.S., F.R.A.S., Honorary Fellow Of Trinity College, Cambridge, Astronomer Royal from 1836 to 1881. Edited by Airy, Wilfrid, BA., M.Inst.C.E. The Project Gutenberg EBook of Autobiography of Sir George Biddell Airy. Release Date: January 9, 2004.Google Scholar
Briggs, H, (1617). Logarithmorum Chilias prima, London.Google Scholar
Briggs, H. (sine data). Henrici Briggii Canon logarithmorum pro numeris serie naturali crescentibus ab 1. ad 20000. Viennae Austriae: typis Joannis Thomae Trattner.Google Scholar
Lapaine, M. (2006): Vektorska analiza, Zbirka riješenih zadataka (Vector Analysis, A Collection of Solved Problems, in Croatian). University of Zagreb, Faculty of Geodesy, Zagreb.Google Scholar
Rambaut, A.A. (1911). Airy, Sir George Biddell, in: Encyclopædia Britannica, (11th ed.).Google Scholar
Snyder, J.P. and Steward, H. (1988). Bibliography of map projections, U. S. Geological Survey Bulletin 1856.Google Scholar
Snyder, J.P. (1993). Flattening the Earth, Two Thousand Years of Map Projections, The University of Chicago Press, Chicago and London.Google Scholar