The Extraordinary Sky of Saddlestania

Question

The infinite country of Saddlestania has some very interesting geography: its elevation from the mathematically flat sea level exactly follows the equation $$\mathbf{z=x^2-y^2}.$$

After traveling along the infinitely long, yet perfectly straight coastline, you arrive at Saddle Point, the only place in the country where you can comfortably take a nap

Drone footage of Saddle Point, Saddlestania, courtesy of Google Maps Calculator.

so you lie down on your back to rest, and look straight up. What do you see?

---

For the more rigorous-minded perspective ponderers, here's the same puzzle without flavour:

If you plot the surface $z=x^2-y^2$ so that every line of sight that intersects the surface is green, and other lines of sight are white, what is the image you get when you place a camera at the origin (x=y=z=0) and point it up (along the positive z axis)?

_{^{This puzzle comes from a friend of mine (T. Lukka), who once forgot to set the camera parameters in the ray tracing software he was testing.}}

Answer 1

You should see

The locus |x|>|y| which is everything vertically between the two main diagonals. Thanks @Bass for providing the picture.

Reason

We need to find all points $x_c,y_c,z_c$ in the camera plane $z = z_c > 0$ such that the ray $tx_c,ty_c,tz_c$ intersects $z=x^2-y^2$ while $t>0$. Solving for $t$ yields $t = z_c/(x_c^2-y_c^2)$ which has a positive solution exactly if $|x_c|>|y_c|$