Note on loxodromes

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

“…Here, T is an orthogonal transformation in E n 1 such that T (e 1 ) = e 1 , T (e 2 ) = e 2 + √ 17) and then, the coefficients of the first fundamental form of M III are given by (2.18) E = ε, F = −cx ′ n (u), and In this section, we obtain the parametrization of the spacelike loxodromes on the spacelike helicoidal surface of type I, type II and type III in a Lorentzian space E n 1 defined by (2.9), (2.12) and (2.16), respectively.…”

“…A curve which cuts all meridians on a rotational surface (or a helicoidal surface) at a constant angle is called as a loxodrome. The equations of the loxodromes on the rotational surfaces in Euclidean 3-space were obtained by Noble [8]. Babaarslan and Munteanu [1] studied time-like loxodromes on the rotational surfaces in Minkowski 3-space.…”

Loxodromes on helicoidal surfaces and tubes with variable radius in E⁴

“…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

“…A rotational surface of the Euclidean three-space is SO(2)-invariant and a loxodrome is a curve on it which meets the meridians at a constant angle. In [13], C.A. Noble obtained the differential equations of a loxodrome on these surfaces and, in particular, he investigated these curves on spheres and spheroids.…”