Differential Equation of a Loxodrome on a Sphere

Abstract

A curve that cuts all meridians of a rotating surface at the same angle is called a loxodrome. If the shape of the Earth is approximated by a sphere, then the loxodrome is a logarithmic spiral that cuts all meridians at the same angle and asymptotically approaches the Earth's poles but never meets them. Since maritime surface navigation defines the course as the angle between the current meridian and the longitudinal direction of the ship, it may be concluded that the loxodrome is the curve of the constant course, which means that whenever navigating on an unchanging course we are navigating according to a loxodrome.

Differential Equation of the Loxodrome on a Helicoidal Surface

“…If we take λ = 0 in Equation (16), we find the general solution of the differential equation of the loxodrome on rotational surface as follows where ε = ± 1. This equation coincides with the equation in (Noble, 1905; Kos et al, 2009).…”

“…Noble (1905) and Kos et al (2009) found the differential equation of the loxodrome on a rotational surface. If we take λ = 0 in Equation (16), we find the general solution of the differential equation of the loxodrome on rotational surface as follows where ε = ± 1.…”

“…Hence they do not measure the shortest distance between two points on the Earth's surface. However loxodromes are primarily used in navigation since they do not require a change of course, i.e., course = θ = constant (Kos et al, 2009). In 1569, a major development for navigation was the Mercator map.…”

“…Noble (1905) found the equations of loxodromes on the rotational surfaces and also gave some particular examples of them such as sphere and spheroid. Kos et al (1999, 2009) computed the arc-length of the loxodrome on a sphere. Also Petrovic (2007) determined the arc-length of the loxodrome on a spheroid.…”

Time-like loxodromes on helicoidal surfaces in Minkowski 3-space

“…Thus, the loxodromes are a popular topic in navigation, see [1] and [12]. From mathematical point of view, geometers have studied loxodromes defined on different kind of surfaces in various ambient spaces and then many works have been found in literature such as [11,15,13,9,5,6,7,10,16].…”