To calculate the loxodromic path, I used the Complex Plane, to draw my map. If the complex number z represents a point on the path, then k*z is another point on the path, for some choice of complex number k. On the Multiplication page you will find details of complex number multiplication. You will also be able to see for yourself how repeated multiplication by a fixed complex number generates a spiral.
We can generate a spiral by repeatedly applying the map T : z|->kz. So the steps along the path are z, kz, (k2)z, (k3)z and so on.
And we can go in the opposite direction with z|->z/k. See the Division page for details on complex division.
However, this just jumps from one point on the path to another, which might not be very close. We can get intermediate points by using powers of which are not whole numbers. For example, k0.1z, k0.2z and so on. See “powers” for how to calculate a complex number raised a power.
The final step is to find a transformation that will map zero and infinity to the points -1 and +1. This is mimics the step in which we twisted the globe round so that the bottom point was on the equator. A suitable transformation is R : z|-> (z-1)/(z+1).
Our spiral which we generated using T : z|->kz assumes that the south pole is at the bottom. But we want to draw the loxodromic path as though a point on the equator is at the bottom. So before we calculate the spiral, we use the inverse transformation R-1 : z |-> (1+z) / (1-z).
And after we have calculated the spiral we transform it back to our new perspective, using R.
So the overall transformation that we need is RTR-1 which works out as
z |-> z(k+1) + (k-1) / z(k-1)+(k+1)
And in this formula we can replace k by kp for any power p, to get smaller steps along the path.
Next Try it for yourself