A New Derivation of the Formula for the Length of a Loxodrome Arc on a Sphere Using Cylindrical Projections

If we take a distance from a map without considering the distortion, we will not get the correct value. We will get the correct value if we take the distortion into account, that is, if we use an expression for distance that does not contain distortion. The article shows how to determine the distance measured along a loxodrome between two points on a sphere if we have an image of the loxodrome in a cylindrical projection. Using the examples of Mercator, equidistant and equal-area projections, a formula was derived that does not depend on the map projection used. This was done with the aim of achieving two goals. The first was that when calculating the distance between two places shown on a map, the distortion inherent in the map due to the applied map projection should be considered. The second goal was to show how, using the theory of map projections, a formula can be derived that does not depend on the applied map projection.