  We determine for each of the simple, simply connected, compact and complex
Lie groups SU(n), Spin$(4n+2)$ and $E_6$ that particular region inside the unit
disk in the complex plane which is filled by their mean eigenvalues. We give
analytical parameterizations for the boundary curves of these so-called trace
figures. The area enclosed by a trace figure turns out to be a rational
multiple of $\pi$ in each case. We calculate also the length of the boundary
curve and determine the radius of the largest circle that is contained in a
trace figure. The discrete center of the corresponding compact complex Lie
group shows up prominently in the form of cusp points of the trace figure
placed symmetrically on the unit circle. For the exceptional Lie groups $G_2$,
$F_4$ and $E_8$ with trivial center we determine the (negative) lower bound on
their mean eigenvalues lying within the real interval $[-1,1]$. We find the
rational boundary values -2/7, -3/13 and -1/31 for $G_2$, $F_4$ and $E_8$,
respectively.
