Table 5: Sample non-linear coordinate combinations, where w, the intermediate coordinate of Eq. (3), has a different meaning for each of these equations It has units given by the spectral coordinate type code as listed in Table 1.



FREQ-W2F
\begin{displaymath}\nu = \left( \frac{\nu_{\rm r}}{\nu_{\rm r}-w} \right) \nu_{\rm r}\end{displaymath} (48)
        

VELO-W2V
\begin{displaymath}v = c~ \frac{A^2_\lambda-1}{A^2_\lambda+1}\end{displaymath} (49)
where
\begin{displaymath}A_\lambda \equiv \frac{c^2-v_{\rm r}^2+cw} {(c-v_{\rm r})\sqrt{c^2-v_{\rm r}^2}}\end{displaymath} (50)
    

VRAD-W2F
\begin{displaymath}V = \frac{V_{\rm r} (c-V_{\rm r}) + cw}{c-V_{\rm r} + w}\end{displaymath} (51)
        

WAVE-F2W
\begin{displaymath}\lambda = \left( \frac{\lambda_{\rm r}} {\lambda_{\rm r}-w} \right) \lambda_{\rm r}\end{displaymath} (52)
        

VELO-F2V
\begin{displaymath}v = c~ \frac{A^2_\nu-1}{A^2_\nu+1}\end{displaymath} (53)
where
\begin{displaymath}A_\nu \equiv \frac{c^2-v_{\rm r}^2-cw}{(c+v_{\rm r}) \sqrt{c^2-v_{\rm r}^2}}\end{displaymath} (54)
    

VOPT-F2W
\begin{displaymath}Z = \frac{Z_{\rm r} (c+Z_{\rm r}) + cw}{c+Z_{\rm r}-w}\end{displaymath} (55)
        

ZOPT-F2W
\begin{displaymath}z = \frac{z_{\rm r} (1+z_{\rm r}) + w}{1+z_{\rm r}-w}\end{displaymath} (56)
        

WAVE-V2W
\begin{displaymath}\lambda = \lambda_0\sqrt{\frac{1+B_\lambda}{1-B_\lambda}}\end{displaymath} (57)
where
\begin{displaymath}B_\lambda \equiv \frac{\lambda_{\rm r}^4-\lambda_0^4 + 4\lambda_0^2\lambda_{\rm r}w}{(\lambda_0^2+\lambda_{\rm r}^2)^2}\end{displaymath} (58)
    

FREQ-V2F
\begin{displaymath}\nu = \nu_0\sqrt{\frac{1-B_\nu}{1+B_\nu}}\end{displaymath} (59)
where
\begin{displaymath}B_\nu \equiv \frac{\nu_0^4-\nu_{\rm r}^4-4\nu_0^2\nu_{\rm r} w} {(\nu_0^2+\nu_{\rm r}^2)^2}\end{displaymath} (60)
    

VOPT-V2W
\begin{displaymath}Z = c \left(\sqrt{\frac{1+D_Z}{1-D_Z}} - 1 \right)\end{displaymath} (61)
where
\begin{displaymath}D_Z \equiv \frac{Y_Z^4 - 1 + 4Y_Zw / c}{(1+Y_Z^2)^2}\end{displaymath} (62)
with
\begin{displaymath}Y_Z \equiv 1 + Z_{\rm r} / c\end{displaymath} (63)


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