Mercator projection
Derivation

As discussed above, the isotropy condition implies that h = k = sec φ. Consider a point on the globe of radius R with longitude λ and latitude φ. If φ is increased by an infinitesimal amount, dφ, the point moves R dφ along a meridian of the globe of radius R, so the corresponding change in y, dy, must be hR dφ = R sec φ dφ. Therefore y′ (φ) = R sec φ. Similarly, increasing λ by dλ moves the point R cos φ dλ along a parallel of the globe, so dx = kR cos φ dλ = R dλ. That is, x′ (λ) = R. Integrating the equations

with x (λ) = 0 and y (0) = 0, gives x(λ) and y(φ). The value λ is the longitude of an arbitrary central meridian that is usually, but not always, that of Greenwich (i.e., zero). The angles λ and φ are expressed in radians. By the integral of the secant function,

The function y (φ) is plotted alongside φ for the case R = 1: it tends to infinity at the poles. The linear y -axis values are not usually shown on printed maps; instead some maps show the non-linear scale of latitude values on the right. More often than not the maps show only a graticule of selected meridians and parallels.
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