26.41 Let $O A B C$ be a unit square in the $x y$ plane, where $O(0,0)$, $A(1,0)$, $B(1,1)$, and $C(0,1)$. Let $u=x^{2}-y^{2}$, $v=2 x y$ be a transformation from the $x y$ plane to the $u v$ plane. The image of the square under this transformation is
(20th American High School Mathematics Examination, 1969)
[Solution] Let $\rightarrow$ denote the given mapping from the $x y$ plane to the $u v$ plane, and let $O^{\prime}, A^{\prime}$, $B^{\prime}, C^{\prime}$ represent the images of $O, A, B, C$ in the $u v$ plane. Direct substitution yields
$$
\begin{array}{l}
O(0,0) \rightarrow O^{\prime}(0,0), \quad A(1,0) \rightarrow A^{\prime}(1,0), \\
B(1,1) \rightarrow B^{\prime}(0,2), \quad C(0,1) \rightarrow C^{\prime}(-1,0) .
\end{array}
$$

Line segment $O A \rightarrow$ line segment $O^{\prime} A^{\prime}$,
Line segment $A B \rightarrow$ parabola $u=1-\frac{v^{2}}{4}$ segment $A^{\prime} B^{\prime}$,
Line segment $B C \rightarrow$ parabola $u=\frac{v^{2}}{4}-1$ segment $B^{\prime} C^{\prime}$,
Line segment $C O \rightarrow$ line segment $C^{\prime} O^{\prime}$.
Note: In $A B$, $x=1$, and in $B C$, $y=1$.
Therefore, the answer is $(D)$.