For example, let the surface $S$ in $\mathbb R^3$ be formed by taking the union of the straight lines joining pairs of points on the lines $\{x=t,y=0,z=1\},\qquad \{x=0,y=1,z=t\}$ with the same parameter $t$.
Then this surface can be given in the parametric form, say $\gamma$: $x=ut,\quad y=1-u,\quad z=u+t-ut.$
I was wondering how to understand $S$ can be given in(?) $\gamma$.
I notice that if we take $u=1$ and $u=0$ in $\gamma$, then respectively, we could get the equations of two original lines. But, does this suffice to show that "$S$ can be given in $\gamma$"? I guess this only shows that the $S$ contains in the curve described by $\gamma$. If so, how to show its converse? Thanks.