Special case of SFL with reference on the Equator

The case where the reference point is chosen on the Equator ( $\delta _0 = 0$ ) is interesting because of the conversion from GLS to SFL suggested by CG02 and described in section 1. Injected in the previous formulas and rules, this gives:

$\displaystyle \phi_p$	$\displaystyle =$	$\displaystyle 0^\circ$	(27)
$\displaystyle \delta_p$	$\displaystyle =$	$\displaystyle +90^\circ$	(28)
$\displaystyle \alpha_p$	$\displaystyle =$	$\displaystyle \alpha_0 - 180^\circ$	(29)

The $\phi_{\text{SFL},0}$ longitude simplifies as:

$\displaystyle \phi_{\text{SFL},0} = \text{atan2}(-\cos\delta\sin(\alpha-\alpha_0-180^\circ), -\cos\delta\cos(\alpha-\alpha_0-180^\circ)) = \alpha-\alpha_0$

(30)

because atan2 $(-y,-x) =$

atan2 $(y,x) + 180^\circ$ (opposite quadrant). Similarly the $\theta_{\text{SFL},0}$ latitude is in this case:

$\displaystyle \theta_{\text{SFL},0} = \sin^{-1}(\sin\delta) = \delta$

(31)

Finally:

$\displaystyle x_{\text{SFL},0}$	$\displaystyle =$	$\displaystyle (\alpha-\alpha_0) \cos \delta$	(32)
$\displaystyle y_{\text{SFL},0}$	$\displaystyle =$	$\displaystyle \delta$	(33)

In this case $(x,y)$

have almost the same definition as the GLS definition provided in Eqs. 1 and 2. If we write a y-shifted version of the GLS equations as:

$\displaystyle x^\prime_$ GLS	$\displaystyle =$	$\displaystyle x_$ GLS $\displaystyle = (\alpha-\alpha_0) \cos \delta$	(34)
$\displaystyle y^\prime_$ GLS	$\displaystyle =$	$\displaystyle y_$ GLS $\displaystyle + \delta_0 = \delta$	(35)

i.e. $(x^\prime_$ GLS $,y^\prime_$ GLS $) = (x_{\text{SFL},0},y_{\text{SFL},0})$ . This shows we can write the GLS projection at any declination as a SFL projection at zero declination with the relationship $x_$

GLS $= x_{\text{SFL},0}$ and $y_$

GLS $= y_{\text{SFL},0} - \delta_0$ .