# 铅垂线

（重定向自铅锤方向

## 曲率

${\displaystyle \kappa ={1 \over g}{\sqrt {g_{x}^{2}+g_{y}^{2}}}}$

### 推导过程

${\displaystyle {\frac {\operatorname {d} \!x}{W_{x}}}={\frac {\operatorname {d} \!y}{W_{y}}}={\frac {\operatorname {d} \!z}{W_{z}}}}$

${\displaystyle \kappa _{1}={\operatorname {d} ^{2}\!x \over \operatorname {d} \!z^{2}}}$

${\displaystyle {\operatorname {d} \!x \over \operatorname {d} \!z}={W_{x} \over W_{z}}\longrightarrow {\operatorname {d} ^{2}\!x \over \operatorname {d} \!z^{2}}={1 \over W_{z}^{2}}\left[W_{z}(W_{xz}+W_{xx}{\operatorname {d} \!x \over \operatorname {d} \!z})-W_{x}(W_{zz}+W_{zx}{\operatorname {d} \!x \over \operatorname {d} \!z})\right]}$

${\displaystyle W_{x}=W_{y}=0\longrightarrow {\operatorname {d} \!x \over \operatorname {d} \!z}=0}$

${\displaystyle \kappa _{1}={W_{zx} \over W_{z}}}$

${\displaystyle \kappa _{1}={g_{x} \over g}}$

${\displaystyle \kappa _{2}={g_{y} \over g}}$

${\displaystyle \kappa ={\sqrt {\kappa _{1}^{2}+\kappa _{2}^{2}}}={1 \over g}{\sqrt {g_{x}^{2}+g_{y}^{2}}}}$

## 参考文献

1. ^ 孔祥元; 郭际明; 刘宗泉. 大地测量学基础. 武汉大学出版社. 2001. ISBN 978-7-30-707562-7.
2. ^ Lu, Zhiping; Qu, Yunying; Qiao, Shubo. Geodesy: Introduction to Geodetic Datum and Geodetic Systems. Springer. 2014-05-23 [2020-04-05]. ISBN 978-3-642-41245-5. （原始内容存档于2020-06-12） （英语）.
3. 现代普通测量学. 清华大学出版社有限公司. 2001: 8 [2020-04-05]. ISBN 978-7-302-04717-9. （原始内容存档于2020-06-12） （中文）.
4. San Francisco W. H. Freeman and Company. Heiskanen Moritz 1967 Physical Geodesy. San Francisco: W. H. Freeman and Company. 1967.
5. ^ 潘正风; 程效军; 成枢; 王腾军; 翟翊. 数字地形测量学. 武汉大学出版社. 2015-07-01. ISBN 978-7-307-15677-7.