| 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$$ x'(\lambda )=R,\qquad y'(\varphi )=R\sec \varphi , $$with $x(λ_0) = 0$ and $y(0) = 0$, gives $x(λ)$ and $y(φ)$. The value $λ_0$ 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,$$ x=R(\lambda -\lambda _{0}),\qquad y=R\ln \left[\tan \left({\frac {\pi }{4}}+{\frac {\varphi }{2}}\right)\right]. $$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. | 考虑地球上半径为 $R$、经度为 $λ$ 和纬度为 $φ$ 的点。如果 $φ$ 增加无穷小的量 $dφ$，则该点沿半径为 $R$ 的地球子午线移动 $R\,dφ$，因此 $y$、$dy$ 的相应变化必须是 $ hR\,dφ = R\sec φ\,dφ$。因此 $y′(φ) = R\sec φ$。类似地，将 $λ$ 增加 $dλ$ 会使点 $R\cos φ\,dλ$ 沿地球平行线移动，因此 $dx = kR\cos φ\,dλ = R\,dλ$。也就是说，$x'(λ) = R$。积分，$$ x'(\lambda )=R,\qquad y'(\varphi )=R\sec \varphi , $$$x(λ_0) = 0$ 和 $y(0) = 0$，给出 $x(λ)$ 和 $y(φ)$。值 $λ_0$ 是任意中央子午线的经度，通常但不总是格林威治的经度（即零）。角度 $λ$ 和 $φ$ 以弧度表示。通过正割函数的积分，$$ x=R(\lambda -\lambda _{0}),\qquad y=R\ln \left[\tan \left({\frac {\pi }{4} }+{\frac {\varphi }{2}}\right)\right]。$$函数 $y(φ)$ 与 $φ$ 一起绘制，用于 $R = 1$ 的情况：它在极点处趋于无穷大。线性 $y$ 轴值通常不会显示在印刷地图上； 相反，一些地图在右侧显示纬度值的非线性比例。通常情况下，地图只显示选定的经线和纬线的标线。 |
