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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!

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