Curve and Surface Fitting

The starting point of this paper is the problem of fitting apolynomial to a set of unequally spaced data. Five alternativeschemes which have been proposed for the solution of this problemwith an automatic computer are mentioned, and the one whichseems best suited to serve as a basis for a general purposeprogram is discussed in some detail. A practical assessmentof the accuracy of the computational processes involved in thismethod is included. The problem in which the data are weightedis also considered, as is the problem in which constraints areimposed upon the fitting function. The procedures describedfor the solution of these two cases are shown to be formallyequivalent. The method favoured for curve fitting is shown to possess stillmore advantages in solving certain types of surface fittingproblems. The general problem, in which data are scattered withina finite plane area, is not discussed in this paper, which concentrateson the case, of great practical importance, in which the dataare situated on parallel straight lines. The most general caseconsidered has a "quasi-rectangular" boundary, composed of apair of parallel sides and a pair of smooth curves. The applicationof constraints is again described; the procedures appropriatefor curve fitting are readily generalized. A practical exampleof the application of the techniques for finding a constrainedsurface is described. Finally there is a brief review of the advantages of these proceduresin common practical situations.