In subdivision schemes for surface generation, the subdivision procedure is applied to the coordinates of vertices in successively finer meshes. The same procedure can be applied to other values associated to the vertices, defining in the limit a continuous function on the subdivision surface. In some applications it seems more natural, though, to associate values not to the vertices but to the faces in successively refined meshes. Values associated to faces in a mesh live on the nodes of the (graph-theoretical) dual of this mesh. Hence a face-based subdivision scheme for quadrilateral or triangle meshes operates on the dual meshes of quadrilateral or triangle meshes with subdivision connectivity. Face-based subdivision schemes are often referred to as vertex-split or dual subdivision schemes, where face-split or primal subdivision schemes are vertex-based.

Subdivision schemes on the dual mesh of a quadrilateral mesh with subdivision connectivity are known---the Doo-Sabin scheme is a well-known example. Because the dual mesh of a regular quadrilateral mesh is again quadrilateral, the analysis of vertex-based and of face-based subdivision schemes in this setting is very similar. In contrast, the dual mesh of a regular triangle mesh consists of hexagons and is not translation-invariant. We analyze subdivision on the hexagonal mesh by reducing it to matrix subdivision on a translation-invariant mesh.