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New Formulae for Combined Spherical Triangles

Published online by Cambridge University Press:  26 October 2018

Tsung-Hsuan Hsieh ,
Shengzheng Wang ,
Wei Liu  and
Jiansen Zhao
Tsung-Hsuan Hsieh*
Affiliation:
(Merchant Marine College, Shanghai Maritime University, Shanghai, China)
Shengzheng Wang
Affiliation:
(Merchant Marine College, Shanghai Maritime University, Shanghai, China)
Wei Liu
Affiliation:
(Merchant Marine College, Shanghai Maritime University, Shanghai, China)
Jiansen Zhao
Affiliation:
(Merchant Marine College, Shanghai Maritime University, Shanghai, China)
*
(E-mail: [email protected])
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Abstract

Spherical trigonometry formulae are widely adopted to solve various navigation problems. However, these formulae only express the relationships between the sides and angles of a single spherical triangle. In fact, many problems may involve different types of spherical shapes. If we can develop the different formulae for specific spherical shapes, it will help us solve these problems directly. Thus, we propose two types of formulae for combined spherical triangles. The first set are the formulae of the divided spherical triangle, and the second set are the formulae of the spherical quadrilateral. By applying the formulae of the divided spherical triangle, waypoints on a great circle track can be obtained directly without finding the initial great circle course angle in advance. By applying the formulae of the spherical quadrilateral, the astronomical vessel position can be yielded directly from two celestial bodies, and the calculation process concept is easier to comprehend. The formulae we propose can not only be directly used to solve corresponding problems, but also expand the spherical trigonometry research field.

Keywords

Spherical trigonometrySpherical triangleWaypointAstronomical vessel position
Type
Research Article
Information
The Journal of Navigation , Volume 72 , Issue 2 , March 2019 , pp. 503 - 512
Copyright
Copyright © The Royal Institute of Navigation 2018 

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References

REFERENCES

Bowditch, N. (2017). The American Practical Navigator, 2017 Edition, Volume I. National Geospatial-Intelligence Agency.Google Scholar
Chen, C.L. (2016). A Systematic Approach for Solving the Great Circle Track Problems Based on Vector Algebra. Polish Maritime Research, 23, 3–13.Google Scholar
Chen, C.L., Hsieh, T.H. and Hsu, T.P. (2015). A Novel Approach to Solve the Great Circle Track Based on Rotation Transformation. Journal of Marine Science and Technology, 23, 13–20.Google Scholar
Chen, C.L., Liu, P.F. and Gong, W.T. (2014). A Simple Approach to Great Circle Sailing: the COFI Method. The Journal of Navigation, 67, 403–418.Google Scholar
Chiesa, A. and Chiesa, R. (1990). A Mathematical Method of Obtaining an Astronomical Vessel Position. The Journal of Navigation, 43, 125–129.Google Scholar
Clough-Smith, J.H. (1978). An Introduction to Spherical Trigonometry. Brown, Son & Ferguson, Ltd.Google Scholar
Cutler, T.J. (2004). Dutton's Nautical Navigation, Fifteenth Edition. Naval Institute Press.Google Scholar
Gery, S.W. (1997). The Direct Fix of Latitude and Longitude from Two Observed Altitudes. NAVIGATION: Journal of The Institute of Navigation, 44, 15–23.Google Scholar
Green, R.M. (1985). Spherical Astronomy. Cambridge University Press.Google Scholar
Holm, R.J. (1972). Great Circle Waypoints for Inertial Equipped Aircraft. NAVIGATION: Journal of The Institute of Navigation, 19(2), 191–194.Google Scholar
Hsu, T.P., Chen, C.L. and Chang, J.R. (2005). New Computational Methods for Solving Problems of the Astronomical Vessel Position. The Journal of Navigation, 58, 315–335.Google Scholar
Karl, J.H. (2007). Celestial Navigation in the GPS Age, Paradise Cay Publications.Google Scholar
MathWorld. (1999). Spherical Trigonometry. http://mathworld.wolfram.com/. Accessed 19 August 2018.Google Scholar
Miller, A.R., Moskowitz, I.S. and Simmen, J. (1991). Traveling on the Curved Earth. Journal of the Institute of Navigation, 38, 71–78.Google Scholar
Murray, D.A. (1908). Spherical Trigonometry: for Colleges and Secondary Schools. Longmans, Green and Co.Google Scholar
Nastro, V. and Tancredi, U. (2010). Great Circle Navigation with Vectorial Methods. The Journal of Navigation, 63, 557–563.Google Scholar
Pepperday, M. (1992). The ‘Two-body Problem’ at Sea. The Journal of Navigation, 45, 138–142.Google Scholar
Pierros, F. (2018). Stand-alone Celestial Navigation Positioning Method. The Journal of Navigation, doi:10.1017/S0373463318000401.Google Scholar
Royal, Navy. (2008). The Admiralty Manual of Navigation: The Principles of Navigation, Volume 1, Tenth Edition. Nautical Institute.Google Scholar
Smart, W.M. (1977). Text-Book on Spherical Astronomy. Cambridge University Press.Google Scholar
Stuart, R.G. (2009). Applications of Complex Analysis to Celestial Navigation. NAVIGATION: Journal of The Institute of Navigation, 56, 221–227.Google Scholar
Tseng, W.K. and Chang, W.J. (2014). Analogues between 2D Linear Equations and Great Circle Sailing. The Journal of Navigation, 67, 101–112.Google Scholar
Todhunter, I. (1986). Spherical Trigonometry: for the Use of Colleges and Schools, Fifth Edition. Macmillan and Company.Google Scholar
Van Allen, J.A. (1981). An Analytical Solution of the Two Star Sight Problem of Celestial Navigation. NAVIGATION: Journal of the Institute of Navigation, 28, 40–43.Google Scholar
Wentworth, G. and Smith, D.E. (1915). Plane and Spherical Trigonometry. Ginn and Company.Google Scholar
Wikipedia. (2001). Spherical Trigonometry. https://en.wikipedia.org/. Accessed 19 August 2018.Google Scholar