We introduce incremental subdivision as a new adaptive subdivision method for triangle meshes. While regular (global) subdivisions produce a smooth surface from a given polygon mesh by refining all of its faces, adaptive subdivision produces a surface by refining only some selected areas of the mesh. Consequently, the selected area becomes fine and high resolution while the rest of mesh is coarse. Incremental subdivision produces a surface whose subdivided area is identical to when the entire mesh is subdivided regularly. In addition, as a good effect, the resolution of the produced surface gradually increases from coarse to fine. The incremental subdivision method expands the specified area to create a buffer region that is subdivided along with it. This method is efficient and easy to implement. We apply the incremental method to Loop and Butterfly subdivision schemes, and we compare it with other adaptive subdivision methods. We discuss some applications of incremental subdivision.