# 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

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 $$|x_c|>|y_c|$$