Ylee15

Hi, this is my Talk page. Thanks. Ylee15 (talk) 15:52, 11 August 2022 (UTC)

Vincenty formula for great-circle distance

In Great-circle distance, the following formula is introduced as a formula that works for *all distances*.

$\Delta \sigma =\arctan {\frac {\sqrt {\left(\cos \phi _{2}\sin(\Delta \lambda )\right)^{2}+\left(\cos \phi _{1}\sin \phi _{2}-\sin \phi _{1}\cos \phi _{2}\cos(\Delta \lambda )\right)^{2}}}{\sin \phi _{1}\sin \phi _{2}+\cos \phi _{1}\cos \phi _{2}\cos(\Delta \lambda )}}.$

But, it seems not. For example, if we have $\Delta \lambda =0$, we have

$\Delta \sigma =\arctan {\frac {|\sin(\theta _{1}-\theta _{2})|}{\cos(\theta _{1}-\theta _{2})}}$

Let's say $\theta _{1}=\pi /2$ and $\theta _{2}=-\pi /2$, then we should have $\Delta \sigma =\pi$. But, the formula above yields $0$. Ylee15 (talk) 16:11, 11 August 2022 (UTC)