approximate). Subdivision surface schemes can also be categorized by the type of polygon that they operate on: some function best for quadrilaterals (quads), while others primarily operate on triangles (tris). Approximating schemes Approximating means that the limit surfaces approximate the initial meshes, and that after subdivision the newly generated control points are not in the limit surfaces. There are five approximating subdivision schemes: Catmull and Clark (1978), Quads – generalizes bi-cubic uniform B-spline knot insertion. For arbitrary initial meshes, this scheme generates limit surfaces that are C2 continuous everywhere except at extraordinary vertices where they are C1 continuous (Peters and Reif 1998). Doo-Sabin (1978), Quads – The second subdivision scheme was developed by Doo and Sabin, who successfully extended Chaikin's corner-cutting method (George Chaikin, 1974) for curves to surfaces. They used the analytical expression of bi-quadratic uniform B-spline surface to generate their subdivision procedure to produce C1 limit surfaces with arbitrary topology for arbitrary initial meshes. An auxiliary point can improve the shape of Doo-Sabin subdivision. After a subdivision, all vertices have valence 4. Loop (1987), Triangles – Loop proposed his subdivision scheme based on a quartic box-spline of six direction vectors to provide a rule to generate C2 continuous limit surfaces everywhere except at extraordinary vertices where they are C1 continuous (Zorin 1997). Mid-Edge subdivision scheme (1997–1999) – The mid-edge subdivision scheme was proposed independently by Peters-Reif (1997) and Habib-Warren (1999). The former used the mid-point of each edge to build the new mesh. The latter used a four-directional box spline to build the scheme. This scheme generates C1 continuous limit surfaces on initial meshes with arbitrary topology. (Mid-Edge subdivision, which could be called "√2 subdivision" since two steps halve distances, could be considered the slowest.) √3 subdivision scheme (2000), Triangles – This scheme was developed by Kobbelt and offers several interesting features: it handles arbitrary triangular meshes, it is C2 continuous everywhere except at extraordinary vertices where it is C1 continuous and it offers a natural adaptive refinement when required. It exhibits at least two specificities: it is a Dual scheme for triangle meshes and it has a slower refinement rate than primal ones. Interpolating schemes After subdivision, the control points of the original mesh and the newly generated control points are interpolated on the limit surface. The earliest work was so-called "butterfly scheme" by Dyn, Levin and Gregory (1990), who extended the four-point interpolatory subdivision scheme for