By the Toeplitz-Hausdorff theorem in convex analysis, the numerical range of a complex square matrix is a convex compact subset of the complex plane. Kippenhahn's theorem describes the numerical range as the convex hull of an algebraic curve that is dual to a hyperbolic curve. For the joint numerical range of several matrices, the direct analogue of Kippenhahn's theorem is known to fail. We discuss the geometry behind these results and prove a generalization of Kippenhahn's theorem that holds in any dimension.