In order to obtain a smooth triangulation of a discrete 3D surface, its smooth representation, typically of continuity, has to be established. This can be achieved by using a suitable subdivision technique. The subdivision is based on the hierarchical recursive refinement of triangular simplices forming the control grid (Fig. 43). Each step of the refinement consists of two stages - splitting and averaging (Fig. 44). In the splitting stage, new nodes are introduced exactly in the middle of individual edges (on the current level of the refinement). During the averaging, the nodes are repositioned to a new location evaluated as a weighted average of nodes in the neighbourhood (according to the averaging mask). As the level of the refinement grows the resulting grid approaches the so called limit surface the smoothness of which depends on the adopted averaging.
The averaging scheme may be either interpolating or approximating. While an interpolating scheme produces the limit surface which is passing through the nodes of the control grid (original as well as refined), the limit surface created by an approximating scheme does not generally interpolate any of the nodes. The averaging scheme may be also classified  as uniform (if the same averaging mask is used everywhere on the surface) or non-uniform (if not), and stationary (if for a given node the same averaging mask is used on each level of the subdivision) or non-stationary (if not).
The most simple triangular subdivision scheme is the Loop's scheme, in which the position of surface node of valence is given by