  Suppose that c is an operator on a Hilbert Space H such that the von Neumann
algebra N generated by c is finite. Suppose that tau is a faithful normal
tracial state on N. Let B denote the spectal scale of c with respect to tau. We
show that the boundary of the numerical range of c is exactly the set of radial
complex slopes on B at the origin. Further, we show that points on this
boundary that lie in the numerical range are visible as line segments in the
boundary of B. Also, line segments on the boundary which lie in the numerical
range show up as faces of dimension two in the boundary of B. Finally, when c
is normal, we prove that the point spectrum of c is exactly the set of complex
slopes of 1-dimensional faces of B.
