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 partial derivative W(A) of its numerical range. We show that (a) if n >= 4 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 partial derivative W(A), and (b) if n >= 3, then partial derivative 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.