Mathematical models of rhumb line and their application

The paper analyses some models of rhumb line on the ellipsoid, sphere and map projection. In geodesy and cartography, rhumb line is a complex curve on the earth's surface that crosses every meridian at the same oblique angle. Also, it is called a loxodrome. Specially, this article described some mathematical models of rhumb line on the different surfaces. So, there are some equations and examples how to calculate distance along rhumb line. In the Mercator projection, rhumb line is displayed as line and it is used in navigation traditionally.