Image of $\mathbb{P}^1 \times \mathbb{P}^1$ by Segre embedding is a hyperboloid

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?

• From the notes: In particular, the "lines" $\{a\}\times\mathbb{P}^1$ and $\mathbb{P}^1\times\{a\}$ in $\mathbb{P}^1\times\mathbb{P}^1$ where the first or second factor is constant, respectively, are mapped to lines in $X \subset \mathbb{P}^3$. The line you suggest is not of the type described above. – Michael Albanese Jul 24 '16 at 23:01
• Dear @MichaelAlbanese Can you please elaborate? I'm not sure where I suggest a specific line. And does it explain why $Im f$ is a hyperboloid? – Emolga Jul 25 '16 at 17:45
• The image of $f$ is the hyperboloid $w_0w_3-w_1w_2=0$ in the three dimensional projective space with coordinates $w_0:w_1:w_2:w_3$. – Georges Elencwajg Jul 26 '16 at 7:18