Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-05T00:49:14.306Z Has data issue: false hasContentIssue false
  • English
  • Français

Application of elliptic integrals in marine navigation

Published online by Cambridge University Press:  15 September 2022

Miljenko Petrović
Miljenko Petrović*
Affiliation:
Professional master mariner, Zagreb, Croatia
*
*Corresponding author. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

If the Earth's oblateness is neglected in marine navigation, then the sphere gives a relatively simple solution for course and distance between any two points. The navigation sphere where a span of one minute of arc is equal to nautical mile is used. The primary deficiency of this approach is the lack of a closed-form formula that takes the Earth's eccentricity into account. Considering the Earth as an oblate spheroid, i.e., a rotational ellipsoid with a small flattening, the problem of computing the length of the meridian arc leads to the understanding of elliptic integrals. In this paper, incomplete elliptic integrals of the first, second and third kind are used to find an arbitrary elliptical arc. The results prove an advantage of using geocentric latitude compared to geodetic and reduced latitude.

Keywords

elliptic integralsmeridian arc lengthmarine navigation
Type
Research Article
Information
The Journal of Navigation , Volume 75 , Issue 5 , September 2022 , pp. 1118 - 1127
Check for updates
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Institute of Navigation

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Carlton Wippern, K. C. (1992). On loxodromic navigation. The Journal of Navigation, 45, 292–297.CrossRefGoogle Scholar
Earle, A. M. (2005). A comment on navigation instruction. The Journal of Navigation, 58, 337–340.CrossRefGoogle Scholar
Hiraiwa, T. (1987). Proposal on the modification of sailing calculations. The Journal of Navigation, 40, 138–148.CrossRefGoogle Scholar
http://www.wolframalpha.com. Accessed 14 March 2022.Google Scholar
Karney, F. F. C. (2011). Geodesics on an Ellipsoid of Revolution. Available at: http://geographiclib.sourceforge.net/geod.html. Accessed 14 March 2022.Google Scholar
Karney, F. F. C. (2012). Algorithms for Geodesics. Available at: http://geographiclib.sourceforge.net/geod.html. Accessed 14 March 2022.Google Scholar
Meyer, T. H. and Rollins, C. (2011). The direct and indirect problem for loxodromes. Journal of the Institute of Navigation, 58, 1–6.CrossRefGoogle Scholar
Petrović, M. (2007). Differential equation of a loxodrome on a spheroid. International Journal of Maritime Science and Technology. ‘Our Sea’, 54, 87–89.Google Scholar
Sadler, D. H. (1956). Spheroidal sailing and the middle latitude. The Journal of Navigation, 9, 371–377.CrossRefGoogle Scholar
Struik, D. J. (1988). Lectures on Classical Differential Geometry. New York: Dover Publications, Inc.Google Scholar
Turner, R. J. (1970). Rhumb-line sailing with a computer. The Journal of Navigation, 23, 233–237.CrossRefGoogle Scholar
Turner, R. J. (1984). A table of latitude parts. The Journal of Navigation, 37, 139.CrossRefGoogle Scholar
Williams, J. E. D. (1950). Loxodromic distances on the terrestrial spheroid. The Journal of Navigation, 3, 133–140.CrossRefGoogle Scholar
Williams, R. (1981). A table of latitude parts. The Journal of Navigation, 34, 247–250.CrossRefGoogle Scholar
Williams, J. E. D. (1982). On a problem of navigation. The Journal of Navigation, 35, 517–519.CrossRefGoogle Scholar
Williams, R. (1982). A table of latitude parts. The Journal of Navigation, 35, 187.CrossRefGoogle Scholar
Williams, R. (1996). Practical rhumb line calculations on the spheroid. The Journal of Navigation, 49, 278.CrossRefGoogle Scholar
Williams, R. (1998). Middle latitude sailing revisited. The Journal of Navigation, 51, 135–140.CrossRefGoogle Scholar