  Suppose that c is a linear operator acting on an n-dimensional complex
Hilbert Space H, and let tau denote the normalized trace on B(H). Set b_1 =
(c+c*)/2 and b_2 = (c-c*)/2i, and write B for the the spectral scale of {b_1,
b_2} with respect to tau. We show that B contains full information about
(W_k)(c), the k-numerical range of c for each k =1,...,n. We then use our
previous work on spectral scales to prove several new facts about (W_k)(c). For
example, we show in Theorem 3.4 that the point lambda is a singular point on
the boundary of (W_k)(c) if and only if lambda is an isolated extreme point of
(W_k)(c). In this case lambda = (n/k)tau(cz), where z is a central projection
in in the algebra generated by b_1, b_2 and the identity. We show in Theorem
3.5, that c is normal if and only if (W_k)(c) is a polygon for each k. Finally,
it is shown in Theorem 5.4 that the boundary of (W_k)(c) is the finite union of
line segments and curved real analytic arcs.
