I have to correct myself on my above quote - it’s plain wrong.
The way to find the “v” component of mapping coordinates in the above example is like shown in the following sketch:
I’ve replaced the “tan(value)” from my previous sketch with their actual results. For the example map covering the -75° … +75° latitude range, the result range has a height of
2 * tan( 75° ) = 7.464.
To arrive at the “v” component at, say, -60°, you take the share of its absolute distance from bottom in relation to the total value range: 2.000 / 7.464 = 0.268.