A number of other procedures exist for interpolation of smooth surfaces from TIN representations. Many of these have their origins in computer-aided geometric design (CAGD), with engineering and visualization applications. Some software packages, such as the groundwater modeling system, GMS and the OpenSource package QGIS, include CAGD interpolation methods, notably Clough-Tocher procedures. These involve fitting a piecewise two-dimensional cubic (i.e. a bicubic) function to each triangular patch, rather as per piecewise cubic spline interpolation on a regular grid. The fitted function may be regarded as being of the form:
The vertices of each triangle provide 3 values, z=f(x,y), and the first-order partial differentials estimated at each vertex zy=fy(x,y) and zx=fx(x,y) provide a further 6, giving a total of 9 values from which the constants in the complete bicubic expression may be determined (Figure 6‑37). However, there are 10 such constants, and at least one additional value is needed to uniquely determine the fitted function. In practice the original triangle is subdivided into three sub-triangles using the centroid of the main triangle. Each sub-triangle includes one edge from the original triangle and the first-order derivative of the Normal to the surface at the midpoint of this edge is used to provide the final parameter. Separate bicubic patches are then fitted to each of the three subtriangles, which collectively provide a continuous spline-like surface approximation at all points. This procedure is exact, fast and smooth, with interpolated values inside the triangle having values that may be above or below the maximum and minimum values of the vertex z-values.