For the two-dimensional (2D) scalar conservation law, when the initial data contain two different constant states and the initial discontinuous curve is a general curve, then complex structures of wave interactions will be generated. In this paper, by proposing and investigating the plus envelope, the minus envelope, and the mixed envelope of 2D non-selfsimilar rarefaction wave surfaces, we obtain and the prove the new structures and classifications of interactions between the 2D non-selfsimilar shock wave and the rarefaction wave. For the cases of the plus envelope and the minus envelope, we get and prove the necessary and sufficient criterion to judge these two envelopes and correspondingly get more general new structures of 2D solutions.