The word loxodromic comes from the Greek loxos meaning oblique and dromos meaning course. Navigating on a fixed (oblique) bearing, results in a path that spirals towards one of the poles (north or south).
A line that follows this sort of spiral is called loxodromic.
To draw a map of this path, I use stereographic projection. This is a method which maps the surface of a sphere onto a plane. For example, it can be used to draw a map of the world on a flat piece of paper.
Imagine that you have a globe made of glass. We draw the loxodromic path on the globe. Stand the globe on the sheet of paper that is to be our map, with the south pole at the bottom. Fix a light to the north pole. The light will shine through the globe, and the path will cast a shadow on the paper.
I drew that a bit wobbly. But never mind, I can get my computer to draw a much nicer map.
Imagine now that we turn the globe round so that the bottom point that rests on the paper is at the equator. Then instead of having the south pole at the origin and the north pole way off at infinity, we get both poles on the map.
Finally, to get my pretty picture, I drew two loxodromic paths, starting at different points. At each step in the journey I drew a straight line between the two paths.
Next The mathematics of the loxodromic path