# The Extraordinary Sky of Saddlestania

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.

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 $$|x_c|>|y_c|$$