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2023 Journal Impact Factor - 0.7
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ISSN 2083-6473
ISSN 2083-6481 (electronic version)
Editor-in-Chief
Associate Editor
Prof. Tomasz Neumann
Published by
TransNav, Faculty of Navigation
Gdynia Maritime University
3, John Paul II Avenue
81-345 Gdynia, POLAND
e-mail transnav@umg.edu.pl
Derivation of Formulas in Spherical Trigonometry Based on Rotation Matrix
1 Shanghai Maritime University, Shanghai, China
2 National Taiwan Ocean University, Zhongzheng, Keelung, Taiwan
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.
KEYWORDS: Astronavigation, Spherical Trigonometry, Rotation Matrix, Spherical Triangle, Formulas of Spherical Triangle, Nautical Mathematics, Geometric Method, Spherical Navigation
REFERENCES
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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
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