This scheme generates C 1 continuous limit surfaces on initial meshes with arbitrary topology. The latter used a four-directional box spline to build the scheme. The former used the mid-point of each edge to build the new mesh. Mid-Edge subdivision scheme (1997–1999) – The mid-edge subdivision scheme was proposed independently by Peters-Reif (1997) and Habib-Warren (1999). Loop (1987), Triangles – Loop proposed his subdivision scheme based on a quartic box-spline of six direction vectors to provide a rule to generate C 2 continuous limit surfaces everywhere except at extraordinary vertices where they are C 1 continuous (Zorin 1997).After a subdivision, all vertices have valence 4. An auxiliary point can improve the shape of Doo-Sabin subdivision. They used the analytical expression of bi-quadratic uniform B-spline surface to generate their subdivision procedure to produce C 1 limit surfaces with arbitrary topology for arbitrary initial meshes. 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.For arbitrary initial meshes, this scheme generates limit surfaces that are C 2 continuous everywhere except at extraordinary vertices where they are C 1 continuous (Peters and Reif 1998). Catmull and Clark (1978), Quads – generalizes bi-cubic uniform B-spline knot insertion.There are five approximating subdivision schemes: 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).Īpproximating means that the limit surfaces approximate the initial meshes, and that after subdivision the newly generated control points are not in the limit surfaces. This is analogous to spline surfaces and curves, where Bézier curves are required to interpolate certain control points, while B-Splines are not (and are more approximate). In general, approximating schemes have greater smoothness, but the user has less overall control of the outcome. Approximating schemes are not they can and will adjust these positions as needed.Interpolating schemes are required to match the original position of vertices in the original mesh.Subdivision surface refinement schemes can be broadly classified into two categories: interpolating and approximating. Mathematically, the neighborhood of an extraordinary vertex (non-4- valent node for quad refined meshes) of a subdivision surface is a spline with a parametrically singular point. In practical use however, this algorithm is only applied a limited, and fairly small ( ≤ 5 ), number of times. The limit subdivision surface is the surface produced from this process being iteratively applied infinitely many times. Each iteration is often called a subdivision level, starting at zero (before any refinement occurs). This resulting mesh can be passed through the same refinement scheme again and again to produce more and more refined meshes. This process produces a denser mesh than the original one, containing more polygonal faces (often by a factor of 4). In many refinement schemes, the positions of old vertices are also altered (possibly based on the positions of new vertices). The positions of the new vertices in the mesh are computed based on the positions of nearby old vertices, edges, and/or faces. This process takes that mesh and subdivides it, creating new vertices and new faces. A refinement scheme is then applied to this mesh. The process starts with a base level polygonal mesh. A subdivision surface algorithm is recursive in nature.
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