## The loxodrome and two projections of the sphere

### Loxodrome

During the Discoveries period, mathematician Pedro Nunes found out that the navigation routes that maintain a steady course, intersecting all meridians with the same angle, determine a curve, which became known as loxodrome*.

Since that time, it is known that the route taken while keeping a constant angle with all meridians is not, in general, the shortest path. The curve that satisfies this property is called great circle arc and it minimizes the distance between two points - for more details, please go to the page on Spherical Geometry). If the Portuguese navigators wanted to follow a great circle arc to follow the shortest path between two places on the planet, they would have to be continuously evaluating and modifying the navigation angle, an impossible task in high sea, at the time. Therefore, the simplest way to navigate through the seas was to keep a constant angle. (Randles, 1989 [1]).

A loxodrome is represented in red, passing through two points; the shortest great circle arc defined by the same two points is displayed in blue (shortest path).

Examples

Definition

* In this work, if there is no ambiguity, we shall identify the curve with its trace, that is, we shall not make a distinction between two curves with the same trace.