The rays are all possible curves $(x(\tau), y(\tau))$. The derivation of the equations is clear and the condition of defining $p_0, q_0$ is $\frac{dx_0}{ds}\frac{dy_0}{d\tau} - \frac{dy_0}{ds}\frac{dx_0}{d\tau} \neq 0$. But:
1) Why is this condition equivalent to "rays not being parallel to $\Gamma$"?
2) What is a similar condition (and why does it hold) when we want the rays to be orthogonal to $\Gamma$?
Any help appreciated!