大圆距离

另一种表达方式是使用出发点和到达点的法矢量与矢量的数量积、向量积和混合积来表达大圆距离[2]

${\displaystyle {\begin{aligned}&\Delta {\hat {\sigma }}={\text{arccos}}\left({\boldsymbol {n}}_{es}^{e}\cdot {\boldsymbol {n}}_{ef}^{e}\right)\\&\Delta {\hat {\sigma }}={\text{arcsin}}\left(\left|{\boldsymbol {n}}_{es}^{e}\times {\boldsymbol {n}}_{ef}^{e}\right|\right)\\&\Delta {\hat {\sigma }}={\text{arctan}}\left({\frac {\left|{\boldsymbol {n}}_{es}^{e}\times {\boldsymbol {n}}_{ef}^{e}\right|}{{\boldsymbol {n}}_{es}^{e}\cdot {\boldsymbol {n}}_{ef}^{e}}}\right)\\\end{aligned}}\,\!}$

此处的${\displaystyle {\boldsymbol {n}}_{es}^{e}\,\!}$ and ${\displaystyle {\boldsymbol {n}}_{ef}^{e}\,\!}$ 分别是起点和终点的n矢量。此处使用的是反正切函数，相对于反余弦函数较为精确，但如果原始数据是以经纬度形式给出，则需要先将经纬度数据转化成n矢量。

链接球面上两点之间的线段就是这两点所在大圆上两点之间的弦，这条弦所对的圆心角可通过几何关系求出，然后再通过弧长公式求出这条弧的弧长，即两点间的大圆距离。[3]

${\displaystyle {\begin{aligned}&\Delta {X}=\cos(\phi _{f})\cos(\lambda _{f})-\cos(\phi _{s})\cos(\lambda _{s});\\&\Delta {Y}=\cos(\phi _{f})\sin(\lambda _{f})-\cos(\phi _{s})\sin(\lambda _{s});\\&\Delta {Z}=\sin(\phi _{f})-\sin(\phi _{s});\\\end{aligned}}\,\!}$

${\displaystyle \mathbb {C} _{h}={\sqrt {(\Delta {X})^{2}+(\Delta {Y})^{2}+(\Delta {Z})^{2}}}}$

圆心角等于：

${\displaystyle \Delta {\widehat {\sigma }}=2\arcsin \left({\frac {C_{h}}{2}}\right).\,\!}$

大圆距离等于：

${\displaystyle d=r\Delta {\widehat {\sigma }}.\,\!}$