A curve in the pseudo-Euclidean plane is circular if it passes through at least one of the absolute points. A cubic can be obtained as a locus of the intersections of a conic and the corresponding line of the projectively linked pencil of conics and pencil of lines. In this paper the conditions that the pencils and the projectivity have to fulfill in order to obtain a circular cubic of a certain type of circularity are determined analytically. The cubics of all types (depending on their position with respect to the absolute figure) can be constructed by using these results. The results are first stated for any projective plane and then their pseudo-Euclidean interpretation is given.