formular is not accurate enough

[miniufo]

Just find out that the formular used to get the lat/lon coordinates from cylindrical coordinates is not accurate enough. The formula are:

$$\begin{align} \phi &= \arcsin(\sin\phi_o\cos\beta + \cos\phi_o\sin\beta\cos\alpha) \\\ \lambda_o-\lambda &= \arcsin(\sin\beta\sin\alpha / \cos\phi) \end{align}$$

where ($\alpha$, $\beta$) are the cylindrical coordinates, ($\lambda$, $\phi$) are the spherical polar coordinates, subscript o indicates the point at the center of the cylindrical coordinate.

The original codes here used a wrong formula:

$$\begin{align} \lambda_o-\lambda &= \arcsin(\sin\beta\sin\alpha) / \cos\phi \end{align}$$

It is interesting that this wrong formula gives approximately the correct results. This can be verified by:

import numpy as np

a1, a2, a3 = np.random.rand(3)*2-1
print(a1, a2, a3)
print('original')
print(np.arcsin(a1*a2/a3))
print(np.arcsin(a1*a2)/a3)
print('reduce a1 or a2')
a1 = a1 / 10
print(np.arcsin(a1*a2/a3))
print(np.arcsin(a1*a2)/a3)

which gives for example:

0.8287575472998463 0.20858890514091422 -0.35868636289313427
original
-0.5028812207377986
-0.48438535604941
reduce a1 or a2
-0.048213882743891996
-0.04819760618048353

This means that when radial angle $\beta$ is small, these two formula are close to each other. This is why I didn't notice this bug here. Now it will be corrected.