New Meridian Arc Formulae by the Least Squares Method

Abstract

Navigational software often lacks official standardisation of the methods used and their accuracy due to commercial confidentiality. The “black box solutions” used by navigational systems are unknown, thus a logical and simple method to solve navigational problems must be presented. This paper presents new meridian arc formulae by the least squares method. As the traditional meridian arc formulae cannot be expressed as a closed form, they are often truncated to the first few terms for practical use and in doing so neglect the values not used. By forming an overdetermined system with known components of the traditional meridian arc formula and actual length of the meridian arc, the least squares method can be used to approximate the best fitting coefficients for the traditional meridian arc formulae and forms the new compact formulae. The new formulae are based on highly accurate values of the meridian arc for the WGS-84 ellipsoid datum, and are perfect for the computational algorithms implemented in navigational software such as Geographic Information Systems (GIS), Electronic Chart Display and Information Systems (ECDIS) and other Electronic Chart Systems (ECS). Their accuracy is compared with other methods and shows that the new proposed formulae are shorter and accurate with negligible errors. The new formulae can be adapted to the accuracy needed and imply different numbers of coefficients. This can also shorten the calculations in navigation such as rhumb-line or great elliptic sailing on the ellipsoid because the meridian arc length is essential for these calculations.