# 大圆距离

## 公式

$\phi_s,\lambda_s;\ \phi_f,\lambda_f\;\!$ 分别代表球面上两点的经纬度，(s代表出发点，f代表前往点)，$\Delta\phi,\Delta\lambda\;\!$ 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:

${\color{white}\Big|}\Delta\widehat{\sigma}=\arccos\big(\sin\phi_s\sin\phi_f+\cos\phi_s\cos\phi_f\cos\Delta\lambda\big).\;\!$

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

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

### 矢量形式

\begin{align} & \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{align}\,\!

### 从弦长求大圆距离

\begin{align} &\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{align}\,\!
$\C_h=\sqrt{(\Delta{X})^2+(\Delta{Y})^2+(\Delta{Z})^2}$

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

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

## 地球上两点间的大圆距离

$R_1 = \frac{2a+b}{3}\,\!$

## 参考文献

1. ^ R.W. Sinnott, "Virtues of the Haversine", Sky and Telescope, vol. 68, no. 2, 1984, p. 159
2. ^ Gade, Kenneth. A non-singular horizontal position representation (PDF). The Journal of Navigation (Cambridge University Press). 2010, 63 (3): 395–417. doi:10.1017/S0373463309990415.
3. ^ http://mathcentral.uregina.ca/QQ/database/QQ.09.09/h/dave2.html
4. ^ McCaw, G. T. Long lines on the Earth. Empire Survey Review. 1932, 1 (6): 259–263.
5. ^ Moritz, H. (1980). Geodetic Reference System 1980, by resolution of the XVII General Assembly of the IUGG in Canberra.
6. ^ Moritz, H. Geodetic Reference System 1980. Journal of Geodesy. 2000-03, 74 (1): 128–133. Bibcode:2000JGeod..74..128.. doi:10.1007/s001900050278.