Loxodromes on helicoidal surfaces and tubes with variable radius in E⁴

Abstract

In this paper, we generalize the equations of loxodromes on helicoidal surfaces and canal surfaces in E 3 to the case of 4-dimension (E 4). Also we give some examples via Mathematica.

Timelike loxodromes on Lorentzian helicoidal surfaces in Minkowski n-space

“…In this section, we determine the parametrization of timelike loxodrome on the timelike helicoidal surface of type I in E n 1 defined by (7). Consider the timelike helicoidal surface of type I, M 1 , in E n 1 given by (7). From a simple calculation, the induced metric g 1 on M 1 is defined by…”

“…Theorem 1. A timelike loxodrome on a timelike helicoidal surface of type I in E n 1 defined by (7) is parametrized by α 1 (u) = H 1 (u, v(u)), where v(u) is given by one of the following functions:…”

“…Helicoidal surfaces are the natural generalizations of rotational surfaces and they play important roles in nature, science and engineering, see [17,18,22]. Thus, this generalization leads the studies to the loxodromes on helicoidal surfaces in [5][6][7][8][9]. Recently, M. Babaarslan and N. Sönmez constructed the three kinds of helicoidal surfaces in E 4 1 by using rotation with 2-dimensional axes and translation in E 4 1 and they also obtained the general form of spacelike and timelike loxodromes on such helicoidal surfaces in [11].…”

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].…”