Appendix

Here we derive the equation of circular rays in constant gradient velocity media. Along a circular ray, the curvilinear abscissa s is, with the previous notations:

$$ds = R d\theta$$ (13)

The equations for the ray trajectory passing through points $S(\rho_s,z_s,\theta_s)$ and $P(\rho,z,\theta)$ are

$$\begin{eqnarray} \rho - \rho_s & = & \int_{\rho_s}^\rho d \rho' = \int_{\theta_... ...r \\ & = & \frac{1}{\gamma p_\rho} ( \sin \theta - \sin \theta_s )\end{eqnarray}$$ (14) (15)

By writing the identity

$$\cos^2 \theta + \sin^2 \theta = \left[ \gamma p_\rho (\rho_... ... \left[ - \gamma p_\rho (z_s - z) + \sin\theta_s \right]^2 = 1$$ (16)

at every point along the ray, we obtain the equation of the desired circle:

$$\left[ \rho - \rho_s - \frac{\cos \theta_s}{\gamma p_\rho} \... ...)}{\gamma} \right]^2 = \left[ \frac{1}{\gamma p_\rho} \right]^2$$ (17)