Differential Equation of the Loxodrome on a Helicoidal Surface

What is it about?

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.

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Photo by Bruno Martins on Unsplash

Photo by Bruno Martins on Unsplash

Why is it important?

Loxodromes correspond to curves that intersect all meridians at a constant angle on a helicoidal surface. Thus, they are very important in navigation. In general, the previous loxodrome studies were associated with rotational surfaces. But in nature, science and engineering, we can find surfaces other than rotational surfaces on which navigation is possible, for example; creeper plants, fractures in geology, parking garage ramps, helicoidal staircases, railways, moving walkways and footbridges, helical channels and so on. For this reason, in the present paper, we investigate the differential equations of loxodromes on the helicoidal surfaces in Euclidean 3-space, hoping that many other researchers work on loxodromes on different structures.