For context, I am attempting to find a method of converting Mercator projections to a globe, as to find the geodesic. Initially I thought Mercator longitude x and latitude y could be converted to coordinates of a globe, longitude λ and latitude φ with the equation $$[x, y] = [λcos(φ), φ]$$ However, when I calculated the metric tensor I found that the latitude vector stretched not by a factor of 1 (which is expected of a globe) but rather, by a variable factor depending on longitude ($(λsin(φ))^2+1$), which led to me figuring out that my original equation referred not to a sphere's surface, but to a sinusoidal map projection. That does bid the question, though: if not $[λcos(φ), φ]$, then what vector does refer to the manifold of a sphere's surface?
P.S. I'm treating Mercator coordinates as Cartesian coordinates, |x| < 180 and |y| < 90.