Approximation of Higher Degree

Abstract

The m wedge basis functions for a polycon of order m and products of these functions may be used to fit discrete data on polycon sides while maintaining continuity across polycon boundaries. The side between vertices i and i + 1 in Fig. 6.1 is a conic boundary component with side node p. Wedges W_i, W_p, and W_i+1 provide a basis for fitting data on this side. The number of basis functions must equal the number of data points. Referring to Sects. 1.3 and 3.4, we recall that on a conic side a linear function has three degrees of freedom and a quadratic function has five degrees of freedom. For each successive degree polynomial there are two more degrees of freedom. In terms of the wedges, a basis for fitting polynomials on the side in Fig. 6.1 is:

$$\displaystyle{ \mathrm{\{W}_{\mathrm{i}}\mathrm{,W}_{\mathrm{i+1}}\mathrm{,W}_{\mathrm{p}}\mathrm{\},}\;\;\{\mathrm{W}_{\mathrm{i}}\mathrm{W}_{\mathrm{p}},\mathrm{W}_{\mathrm{p}}\mathrm{W}_{\mathrm{i+1}}\},\;\;\{\mathrm{W}_{\mathrm{i}}\mathrm{W}_{\mathrm{p}}\mathrm{W}_{\mathrm{i+1}},\mathrm{W}_{\mathrm{i}}^{\mathrm{2}}\mathrm{W}_{\mathrm{ p}}\},\;.\;.\;. }$$

(6.1)

where the first set suffices for linear fitting, the second set is appended for quadratic fitting, the third for cubic fitting, etc. Wedges W_i, W_p, and W_i+1 differ in the two polycons that share this boundary component. These functions, however, coincide on the boundary. If the basis in Eq. (6.1) is used in approximating data over both polycons, then continuity will be achieved across this boundary. The continuations into the respective polycons will of course differ in general. In Fig. 6.1b, for example, the six data points indicated by X may be fit by the five functions in the first two sets in (6.1) and any one of the functions in the third set.