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By establishing a coordinate system, introducing the concept of rotation floating axis, dividing the first/second half period on the rise, especially introducing offset, under the premise framework of the 1-D synthesis on the scheduled bearing line of linkage, the existing form and solution concept of the whole nested interval, the existing form of roller center and theory base circle radius, and the solution theory of inferior/non inferior solution nested region and the optimal solution of interval were presented. Accordingly, the solution set of offset mechanism were solved. Under the premise framework of the 2-D synthesis on the non-scheduled bearing line of linkage, the proposed concept and its solution theory of whole mechanism solution, inferior/non-inferior solution nested region and the optimal solution of the whole solution were presented. Based on the research result of the class Ⅱ synthesis problem of the non-offset mechanism, the 1-D and 2-D class Ⅱ synthesis of offset mechanism were solved. It enriched the solution space and optimal solution space, and demonstrated the important theoretical value and practical significance of the offset mechanism. Finally, by providing two synthesis examples, it demonstrated that the offset mechanism was greatly superior to the non-offset mechanism. 