Vector Solutions for Great Circle Navigation

Abstract

Traditionally, navigation has been taught with methods employing Napier's rules for spherical triangles while methods derived from vector analysis and calculus appear to have been avoided in the teaching environment. In this document, vector methods are described that allow distance and azimuth at any point on a great circle to be determined. These methods are direct and avoid reliance on the formulae of spherical trigonometry. The vector approach presented here allows waypoints to be established without the need to either ascertain the position of the vertex or select the nearest pole; the method discussed here requires only one spherical triangle having an apex at the North Pole and is also easy to implement on a small computer.

Analogues between 2D Linear Equations and Great Circle Sailing

“…This Latitude Equation of Longitude is consistent with the equations of Chen et al (2004), and Earle (2005). This Latitude Equation of Longitude is consistent with the equations of Chen et al (2004), and Earle (2005).…”

“…PA R A M E T R I C E Q U AT I O N S U S E D FO R G R E AT C I R C L E S A I L I N G . The great circle is more than enough for practical uses and has the advantage of being much less complicated than the great ellipse and geodesic on a spheroid (Clynch, 2013;Earle, 2005Earle, , 2006. If the departure and destination points were antipodal then there would be infinite arcs that would suffice as the shortest distance for the great circle sailing.…”

“…The smaller portion of a great circle is the shortest path between two points on the sphere; this path is known as the geodesic on a sphere, while travelling on this path is called great circle sailing, or great circle navigation. Some works, such as those by Chen, Hsu and Chang (2004) and Earle (2005Earle ( , 2006 approach the great circle with interesting methods. They can also be found online at Wolfram MathWorld (2013).…”

Great Circle Navigation with Vectorial Methods

“…This problem has been already examined in a previous paper of the first author (Nastro, 2000), and recently discussed in this Journal (Earle, 2005;Tseng and Lee, 2007); the present manuscript reports some of this previous paper's results in a more compact form.…”

Building the Latitude Equation of the Mid-longitude

“…An equation for the geodesic on spherical surface, namely the great circle equation [1] [5] [10], could be found in many textbooks of variation calculus [4] [6] [11] and a well-known mathematics website [12]. Their great circle equation is ingenious and creative.…”