From projection plane to celestial coordinates

$(x,y)$ projection planes coordinates can be converted to the absolute celestial coordinates $(\alpha ,\delta )$ as follows. The CG02 equations 92 and 93 for the SFL projection are written:

$$\phi = \frac{x}{\cos y}$$ (19)

$$\theta = y$$ (20)

Using their eq. 2 with the parameters computed before, the coordinates are:

$$\alpha = \alpha_p + (-\cos\theta\sin(\phi-\phi_p), \sin\theta\cos\delta_p-\cos\theta\sin\delta_p\cos(\phi-\phi_p))$$ (21)

$$\delta = \sin^{-1}(\sin\theta\sin\delta_p+\cos\theta\cos\delta_p\cos(\phi-\phi_p))$$ (22)