已知经纬度求距离

如下图所示，已知点A的经度$\theta_A$θA​和纬度$\varphi_A$φA​以及点B的经纬度$\theta_B$θB​和$\varphi_B$φB​，求两点的距离。图中点C为地球北极，其与点A和点B构成了一个球面三角形$\bigtriangleup ABC$△ABC，三角形三边长分别为$a$a、$b$b、$c$c。点F、D、E分别为本初子午线、过点A的经线以及过点B的经线与赤道面的交点。
首先将地球视为一单位圆球，则根据球面三角形余弦定理，有$cosc=cosacosb+sinasinbcosC$cosc=cosacosb+sinasinbcosC
对于单位圆球来说，球面三角形边的弧长即为相应球心角的弧度，即$b=\angle AOC=\pi/2-\varphi_A$b=∠AOC=π/2−φA​ $a=\angle COB=\pi/2-\varphi_B$a=∠COB=π/2−φB​
$\angle C$∠C是平面$COD$COD和平面$COE$COE的夹角$\angle C=\angle DOE=\theta_B-\theta_A$∠C=∠DOE=θB​−θA​
代入余弦定理公式，得$cosc=cos(\pi/2-\varphi_A)cos(\pi/2-\varphi_B)+sin(\pi/2-\varphi_A)sin(\pi/2-\varphi_B)cos(\theta_B-\theta_A)\\=sin(\varphi_A)sin(\varphi_B)+cos(\varphi_A)cos(\varphi_B)cos(\theta_B-\theta_A)$cosc=cos(π/2−φA​)cos(π/2−φB​)+sin(π/2−φA​)sin(π/2−φB​)cos(θB​−θA​)=sin(φA​)sin(φB​)+cos(φA​)cos(φB​)cos(θB​−θA​)
因此$c=arccos(sin(\varphi_A)sin(\varphi_B)+cos(\varphi_A)cos(\varphi_B)cos(\theta_B-\theta_A))$c=arccos(sin(φA​)sin(φB​)+cos(φA​)cos(φB​)cos(θB​−θA​))
再由弧长计算公式$弧长=弧度\times半径$弧长=弧度×半径
则可计算两点间得距离$S=c\times R_e=R_e\times arccos(sin(\varphi_A)sin(\varphi_B)+cos(\varphi_A)cos(\varphi_B)cos(\theta_B-\theta_A))$S=c×Re​=Re​×arccos(sin(φA​)sin(φB​)+cos(φA​)cos(φB​)cos(θB​−θA​))
其中$R_e为地球半径$Re​为地球半径