Indefinite numerical range of $3\times 3$ matrices

summary:The point equation of the associated curve of the indefinite numerical range is derived, following Fiedler's approach for definite inner product spaces. The classification of the associated curve is presented in the $3\times 3$ indefinite case, using Newton's classification of cubic curves. Illustrative examples of all the different possibilities are given. The results obtained extend to Krein spaces results of Kippenhahn on the classical numerical range