I have tried to reproduce the transformation rules when one does polar stereographic projection from Mathworld web. Similar equations appear in Snyder (1987).
When I do the calculation, I get (for the projection centered on the north pole):
$$ \begin{align} x' &= 2R \tan{\frac{\theta}{2}}\cos{\phi}\,\\ y' &= 2R \tan{\frac{\theta}{2}}\sin{\phi}\, \end{align} $$
where the angles and symbols are defined below in the image. $x'$ and $y'$ axes have the same direction as the $x$ and $y$ axes. The formula from the above reference gives:
$$ \begin{align} x' &= 2R \frac{\cos{\theta_L}}{1+\sin{\theta_L}}\sin{(\lambda-\lambda_0)}\,\\ y' &= -2R \frac{\cos{\theta_L}}{1+\sin{\theta_L}}\cos{(\lambda-\lambda_0)}\,, \end{align} $$ where according the the Mathworld "$\lambda_0$ is the central longitude" (whatever that is supposed to be) and $\theta_L = \frac\pi2 - \theta$ and with a little algebra, one can show that:
$$ \begin{align} \tan{\frac\theta2} &= \frac{\cos{\theta_L}}{1+\sin{\theta_L}}\,. \end{align} $$
So the only difference is in the last terms. They suggest the relationship:
$$ \begin{align} \phi = \lambda - \lambda_0 - \frac\pi2\,. \end{align} $$
This suggests that my solution and projection is somehow rotated along the z-axis. I do not really understand the concept of $\lambda_0$ in the first place. When one browses the catalogue file DSMAP_POLAR.CAT for the LOLA lunar mission, they use the above Mathworld equation with $\lambda_0=0$. Can someone explain why all the references have the $\sin$ instead of $\cos$ and $-\cos$ instead of $\sin$, that is why is my solution unusual (why is the common practice different)?
