# 大圆距离

## 公式

${\displaystyle \phi _{s},\lambda _{s};\ \phi _{f},\lambda _{f}\;\!}$ 分别代表球面上两点的纬度和经度，(s代表出发点，f代表前往点)，${\displaystyle \Delta \phi ,\Delta \lambda \;\!}$ 是两者差的绝对值，那么两点之间的圆心角可由球面余弦定律所给出:

${\displaystyle {\color {white}{\Big |}}\Delta {\widehat {\sigma }}=\arccos {\big (}\sin \phi _{s}\sin \phi _{f}+\cos \phi _{s}\cos \phi _{f}\cos \Delta \lambda {\big )}.\;\!}$

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

${\displaystyle {\color {white}{\frac {\bigg |}{|}}}\Delta {\widehat {\sigma }}=2\arcsin \left({\sqrt {\sin ^{2}\left({\frac {\Delta \phi }{2}}\right)+\cos {\phi _{s}}\cos {\phi _{f}}\sin ^{2}\left({\frac {\Delta \lambda }{2}}\right)}}\right).\;\!}$

### 矢量形式

{\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}}\,\!}

### 从弦长求大圆距离

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

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

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

## 地球上两点间的大圆距离

${\displaystyle R_{1}={\frac {2a+b}{3}}\,\!}$

## 参考文献

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3. ^ A tunnel from Toronto to Montreal. Math Central. [2013-05-24]. （原始内容存档于2020-07-17）.
4. ^ McCaw, G. T. Long lines on the Earth. Empire Survey Review. 1932, 1 (6): 259–263.
5. ^ Moritz, H. Geodetic reference system 1980. Bulletin Géodésique. 1980-09, 54 (3): 395–405. ISSN 0007-4632. doi:10.1007/bf02521480.
6. ^ Moritz, H. Geodetic Reference System 1980. Journal of Geodesy. 2000-03, 74 (1): 128–133. Bibcode:2000JGeod..74..128.. ISSN 0949-7714. doi:10.1007/s001900050278 （英语）.