# 参心大地坐标与参心空间直角坐标的转换

## 参心大地坐标转换为参心空间直角坐标

$$\begin{cases} & X=(N+H)\cos{B}\cos{L} \\ & Y=(N+H)\cos{B}\sin{L} \\ & Z=[ N(1- \mathit{e}^2)+H) ]\sin{B} \end{cases}$$公式中，$N$ 为椭球面卯酉圈的曲率半径，$\mathit{e}$ 为椭球的第一偏心率：$$\mathit{e}^2 = \frac{a^2-b^2}{a^2}$$ $a$ ，$b$ 椭球的长短半径 $$N=\frac{a}{\sqrt{1-\mathit{e}^2\sin^2B}}$$

## 参心空间直角坐标转换为参心大地坐标

$$\begin{cases} & L=\arctan\left(\frac{Y}{X}\right) \\ & B=\arctan\left(\frac{Z\left(N+H\right)}{\sqrt{\left(X^2+Y^2 \right)\left[N\left(1-\mathit{e}^2 \right) + H \right]}}\right) \\ & H=\frac{Z}{\sin B}-N\left(1-\mathit{e}^2 \right) = \frac{\sqrt{X^2 + Y^2}}{\cos B} - N \end{cases} $$

大地经度 $L$ ，$B$ 求解要考虑 $Y$ 和 $Z$ 的正负。

大地纬度 $B$ 的计算比较复杂，变换后 $$\tan B = \frac{Z+N\mathit{e}^2\sin B}{\sqrt{X^2 + Y^2}}$$ 上式中右端有待定量 $B$ ，需要迭代计算。迭代时可取 $$\tan B_1=\frac{Z}{\sqrt{X^2+Y^2}}$$ 用 $B$ 的初值 $B_1$ 计算 $N_1$ 和 $\sin B_1$ ，然后继续迭代，直至最后两次 $B$ 值之差小于允许误差为止。

由于左、右两端具有不同的三角函数，这对于迭代很不方便。为克服这一缺点，建议采用下面的迭代公式：

$$N=\frac{c}{V}=\frac{c}{\sqrt{1+{e'}^2\cos^2 B}}$$

$$\frac{1}{\cos^2 B}=1+\tan^2 B$$

整理得 $$\tan B = \frac{Z}{\sqrt{X^2+Y^2}}+\frac{ce^2\tan B}{\sqrt{X^2+Y^2} \cdot \sqrt{1+{e'}^2+\tan^2 B}}$$ 因此，$$t_{i+1}=t_0+\frac{pt_i}{\sqrt{k+t_i^2}}$$ 式中 $$\begin{cases} & t_0=\frac{Z}{\sqrt{X^2+Y^2}} \\ & p=\frac{ce^2}{\sqrt{X^2+Y^2}} \\ & k=1+{e'}^2 \\ & c=a\sqrt{1+{e'}^2} \\ & {e'}^2 = \frac{a^2-b^2}{b^2} \end{cases}$$ $t_i$ 为前一次迭代值，第一次迭代令 $t_i=t_0$ 。

参考资料:

《大地测量基准与坐标转换》，姚宜斌，武汉大学测绘学院卫星应用工程研究所，2006.03. PDF链接

《大地测量学基础》，孔祥元 郭际明 刘宗泉，武汉大学出版社, P103.

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