Classification of joint numerical ranges of three hermitian matrices of size three

The joint numerical range W(F) of three hermitian 3-by-3 matrices F=(F1,F2,F3) is a convex and compact subset in R3. Generically we find that W(F) is a three-dimensional oval. Assuming dim⁡(W(F))=3, every one- or two-dimensional face of W(F) is a segment or a filled ellipse. We prove that only ten configurations of these segments and ellipses are possible. We identify a triple F for each class and illustrate W(F) using random matrices and dual varieties.