So I have been going through the internet and stumbled upon Rhumb lines on spheres in $R^3$ and did read that they are straight lines on the Mercator projection of that said sphere.
At first I thought that is because the Mercator projection preserves angles (conformal mapping). But there are plenty of conformal map projections like the Stereographic projection which do not project Rhumb lines into straight ones.
After doing some additional research I found out that this property is dependant on the preservation of Axes, i.e. the direction to lets say the north pole is the same at any point of the projection.
Now my Question is the following: How would someone proof that Rhumb lines become straight lines after a projection with those two properties?
Also: Are those two statements equivalent?