Journal is indexed in following databases:

2022 Journal Impact Factor - 0.6
2022 CiteScore - 1.7




ISSN 2083-6473
ISSN 2083-6481 (electronic version)




Associate Editor
Prof. Tomasz Neumann

Published by
TransNav, Faculty of Navigation
Gdynia Maritime University
3, John Paul II Avenue
81-345 Gdynia, POLAND
Derivation of Formulas in Spherical Trigonometry Based on Rotation Matrix
1 Shanghai Maritime University, Shanghai, China
2 National Taiwan Ocean University, Zhongzheng, Keelung, Taiwan
ABSTRACT: The formulas of spherical triangle, which are widely used to solve various navigation problems, are the important basic knowledge of nautical mathematics. Because the sine rules and the cosine rules for the sides are the fundamental formulas to derive the other spherical triangle formulas, they are also called the genetic codes of the spherical triangle formulas. In the teaching process, teachers usually use the geometric method to derive and prove these fundamental formulas. However, the derivation of geometric methods is complicated and difficult to understand. To improve the teaching process, this paper proposes the three-dimensional rotation method, which is based on conversion of two cartesian coordinate frames using the rotation matrices. This method can easily and simultaneously derive the sine rules, the cosine rules for the sides, and the five-part formulas (I), and is also helpful to solve different kinds of spherical navigation problems.
Hsu T. P., Chen C. L., Chang J. R. (2005). New Computational Methods for Solving Problems of the Astronomical Vessel Position. The Journal of Navigation, 58(2): 315-335. - doi:10.1017/S0373463305003188
Clough-Smith J. H. (1966). An Introduction to Spherical Trigonometry. Brown, Son & Ferguson, Ltd.
Todhunter I. (1886). Spherical Trigonometry: For the Use of Colleges and Schools”, Fifth ed., Macmillan and Company.
Smart W. M. (1977). Text-Book on Spherical Astronomy, Sixth ed., Cambridge University Press.
Green R. M. (1985). Spherical Astronomy, Cambridge University Press.
Arfken G. (1985). Mathematical Methods for Physicists, Third ed., Academic Press.
Bowditch N. (1981). American Practical Navigator, Pub. 9, Vol. 2, Defense Mapping Agency Hydrographic/Topographic Center.
Citation note:
Hsieh T.H., Wang S.Z., Liu W., Zhao J.S., Chen C.L.: Derivation of Formulas in Spherical Trigonometry Based on Rotation Matrix. TransNav, the International Journal on Marine Navigation and Safety of Sea Transportation, Vol. 13, No. 3, doi:10.12716/1001.13.03.09, pp. 553-558, 2019

Other publications of authors:

File downloaded 822 times

Important: cookie usage
The website uses certain cookies. A cookie is a text-only string of information that the TransNav.EU website transfers to the cookie file of the browser on your computer. Cookies allow the website to perform properly and remember your browsing history. Cookies also help a website to arrange content to match your preferred interests more quickly. Cookies alone cannot be used to identify you.
Akceptuję pliki cookies z tej strony