Loxodromes on Twisted Surfaces in Euclidean 3-Space

Abstract

In the present paper, loxodromes, which cut all meridians and parallels of twisted surfaces (that can be considered as a generalization of rotational surfaces) at a constant angle, have been studied in Euclidean 3-space and some examples have been constructed to visualize and support our theory.

Loxodromes on Twisted Surfaces in Lorentz-Minkowski 3-Space

“…For example, the differential equations of loxodromes on a sphere in Euclidean 3-space have been given in [15] and since spheroid model of loxodrome calculation may prove useful for understanding mathematics of navigation, the differential equations of a loxodrome on the spheroid has been obtained in [17]. Also, loxodromes on a twisted surface, on a canal surface, on a helicoidal surface and on a rotational surface in E 3 have been studied in [2], [3], [8] and [16], respectively. Furthermore, spacelike or timelike loxodromes on different surfaces in Lorentz-Minkowski space have been studied in [5], [6], [9], [21] and etc.…”

“…= ((a + f (y 0 ) cosh(bx) + g(y 0 ) sinh(bx)) cos x, 2 and so, the meridian is always spacelike. Hence, for the cases of our "surface is spacelike, loxodrome is spacelike, meridian is spacelike", "surface is timelike, loxodrome is spacelike, meridian is spacelike" and "surface is timelike, loxodrome is timelike, meridian is spacelike" we can take b = 0; but for the case of our "surface is timelike, loxodrome is spacelike, meridian is timelike" and "surface is timelike, loxodrome is timelike, meridian is timelike" we cannot take b = 0.…”

“…= ((a + f (y 0 ) cos(bx) − g(y 0 ) sin(bx)) cosh x, f (y 0 ) sin(bx) + g(y 0 ) cos(bx), (a + f (y 0 ) cos(bx) − g(y 0 ) sin(bx)) sinh x), 2 and hence, the meridian is always timelike. So, for the cases of our "surface is timelike, loxodrome is spacelike, meridian is timelike" and "surface is timelike, loxodrome is timelike, meridian is timelike" we can take b = 0; but for the case of our "surface is spacelike, loxodrome is spacelike, meridian is spacelike", "surface is timelike, loxodrome is spacelike, meridian is spacelike" and "surface is timelike, loxodrome is timelike, meridian is spacelike" we cannot take b = 0.…”