Let $f: \mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^3$ be the Segre embedding given by $((x_0:x_1),(y_0:y_1))\mapsto(x_0y_0,x_0y_1,x_1y_0,x_1y_1)$. Gathmann's notes claim that the real points of the image of $f$ form a hyperboloid, and that lines map to lines as in this picture:
I'm having trouble seeing why these two facts holds. Using the affine coordinates on each $\mathbb P^1$, we get that $f$ sends $((1:x),(1:y))$ to $(1:y:x:xy)$, so as an affine function we have $(x,y) \mapsto (y,x,xy)$. This seems to me as a mirror image of the surface $z=xy$, which is a hyperbolic paraboloid and looks very different from a one-sheeted hyperboloid. What am I missing? And how can we see the geometric phenomenon that is pictured above?
As mentioned by Georges Elencwajg, the image of this map is $w_0w_3 - w_1w_2=0$. Making the change of coordinates $$ w_0 = x_0+x_3\\ w_3 = x_0-x_3\\ w_1 = x_1+x_2\\ w_2 = x_1-x_2$$ we find $$x_0^2-x_1^2+x_2^2-x_3^2=0$$ Finally, taking the chart $x_3=1$, one gets the hyperboloid.