Differential Equation of the Loxodrome on a Helicoidal Surface

Abstract

In nature, science and engineering, we often come across helicoidal surfaces. A curve on a helicoidal surface in Euclidean 3-space is called a loxodrome if the curve intersects all meridians at a constant azimuth angle. Thus loxodromes are important in navigation. In this paper, we find the differential equation of the loxodrome on a helicoidal surface in Euclidean 3-space. Also we give some examples and draw the corresponding pictures via the Mathematica computer program to aid understanding of the mathematics of navigation.

Loxodromes on helicoidal surfaces and tubes with variable radius in E⁴

“…The orbit of a plane curve under a screw motion is called as helicoidal surface and it is a natural generalization of rotational surface. The equations of loxodromes on helicoidal surfaces in E 3 were found by Babaarslan and Yayli [3].…”

“…Loxodromes don't need a change of course and thus, they are usually used in navigation. Noble [11] investigated the equations of loxodromes on the rotational surfaces in Euclidean 3-space E 3 . The orbit of a plane curve under a screw motion is called as helicoidal surface and it is a natural generalization of rotational surface.…”

Loxodromes on Invariant Surfaces in Three-Manifolds

“…The corresponding surface is the twisted sphere (see the Figure 1). 1 The elliptic integral of the second kind is defined by Example 2.9 (Twisted pseudosphere). If we consider a = 1 and ξ 1 (u) = e −u , the integration of (17) leads to the Gaussian hypergeometric function 2 and…”

“…Let M be a G X -invariant surface of (H 2 × R, g), and letγ(u) = (ξ 1 (u), ξ 2 (u)) be its profile curve in the regular part of the orbit space (B = H 2 × R/G X ,g), which is parametrized by the invariant functions ξ 1 and ξ 2 . With respect to the local parametrization ψ(u, v) given by (1) we have:…”

Space-like loxodromes on the canal surfaces in Minkowski 3-space

“…In this section we give necessary concepts related to curves and surfaces in Minkowski 3-space E 3 1 . The Lorentzian scalar product of the vectors u = (u 1 , u 2 , u 3…”