  Let $\mu_1$ be a complex number in the numerical range $W(A)$ of a normal
matrix $A$. In the case when no eigenvalues of $A$ lie in the interior of
$W(A)$, we identify the smallest convex region containing all possible complex
numbers $\mu_2$ for which $\begin{bmatrix}\mu_1& *\\0& \mu_2\end{bmatrix}$ is a
$2$-by-$2$ compression of $A$.
