Line segments and elliptic arcs on the boundary of a numerical range

For an n-by-n complex matrix A, we consider the numbers of line segments and elliptic arcs on the boundary W(A) of its numerical range. We show that (a) if [image omitted] and A has an (n - 1)-by-(n - 1) submatrix B with W(B) an elliptic disc, then there can be at most 2n - 2 line segments on W(A), and (b) if [image omitted], then W(A) contains at most (n - 2) arcs of any ellipse. Moreover, both upper bounds are sharp. For nilpotent matrices, we also obtain analogous results with sharper bounds.