the right coordinate makes a structure obvious — a change of chart that trivializes what looked hard and reveals the invariant that was always there

The Mercator projection is the complex logarithm of stereographic projection. Stereographic projection sends the sphere to the complex plane conformally; the complex logarithm sends ℂ* to a horizontal strip, turning multiplication into addition and rotation into vertical translation. Composed, they are Mercator — which is why Mercator is conformal, and why a loxodrome (a path of constant compass bearing) becomes a straight line: a constant bearing is a constant argument increment, and log turns that into a constant slope. None of this is visible in the sphere's own coordinates; the right chart makes the conformal structure and the straight-line rhumb both immediate.

Calibration question: when a structure looks hard in the coordinates it is handed in, ask whether a single change of chart — often a familiar one in disguise, here log ∘ stereographic — trivializes it. Suspect that the difficulty is in the representation, not the object, and hunt the chart that makes the invariant obvious.