6. Let quadrilateral $ABCD$ be a square on the $xy$ plane, with vertices $A(0,0)$, $B(1,0)$, $C(1,1)$, and $D(0,1)$. Given the transformation $u=x^{2}-y^{2}$, $v=2xy$ from the $xy$ plane to the $uv$ plane, the image set of the square $ABCD$ is ( ).

Figure 1 is one of the options.
6. C.
$$
\begin{array}{l}
\text { Line segment } A B:\left\{\begin{array}{l}
0 \leqslant x \leqslant 1, \\
y=0
\end{array}\right. \\
\Rightarrow\left\{\begin{array} { l } 
{ u = x ^ { 2 } ( 0 \leqslant u \leqslant 1 ) , } \\
{ v = 0 }
\end{array} \Rightarrow \left\{\begin{array}{l}
0 \leqslant u \leqslant 1, \\
v=0 .
\end{array}\right.\right.
\end{array}
$$

Line segment $B C$ :
$$
\begin{array}{l}
\left\{\begin{array} { l } 
{ x = 1 , } \\
{ 0 \leqslant y \leqslant 1 }
\end{array} \Rightarrow \left\{\begin{array}{l}
u=1-y^{2}(0 \leqslant u \leqslant 1), \\
v=2 y
\end{array}\right.\right. \\
\Rightarrow v^{2}=-4(u-1)(0 \leqslant u \leqslant 1,0 \leqslant v \leqslant 2) .
\end{array}
$$