Cubic precision Clough-Tocher interpolation

The standard Clough-Tocher split-domain scheme constructs a surface element with quadratic precision. In this paper, I will look at methods for improving the degrees of freedom in Clough-Tocher schemes. In particular , I will discuss modiications to the cross-boundary construction that improve the interpolant from quadratic precision to cubic precision. In the general scattered data interpolation problem, we are trying to nd a smooth (at least C 1 continuous) bi-variate function F (x; y) such that F interpolates a set of data values at prescribed locations, i. In triangular scattered data tting, we also have a set of triangles T = fT 0 ; : : : ; T n?1 g that form a proper triangulation 9] with the vertices of T 2 T being from f(x i ; y i)g. Commonly, we will also have normals ((rst partial derivatives) at the data points. We could try to t a single cubic patch per triangle, which we will express in B ezier form as in Figure 1 (see Farin's book 5] for details on triangular B ezier patches). For the patch to interpolate the data points and normals, the V i and the T ij are uniquely determined. This leaves us a single center control point. Unfortunately, the single degree of freedom in this control point is inadequate