Data Fitting with B-splines Curves

Abstract

During CAD processes, we are often confronted with the problem of defining a curve from some ordered points. The curve must be included in the data base and must be defined with the implemented model. Our work concerns the B-splines model because it is the most often encountered and we must notice that the NURBS model is often used with equal weights, like the B-splines model. The curves we obtain can be transformed into NURBS curves, using for example a weight optimization process [1]. The first technique for solving this problem is to take the given points as control points, which is not of interest for this paper. The points may be intented to lie on the curve or near the curve. It is then a problem of interpolation or fitting. Interpolation is known to be a particular case of fitting. Separate algorithms correspond to these two approaches [2,3,4]. We introduce a single form, as general as possible, to solve both problems. This involves the solving of a linear system (the problem is not a linear one for NURBS in the general case but interesting approaches are studied in [5,6,7]). We present our method in two dimensions. The latter can easily be extended in space, for curves or surfaces. In a CAGD application, the control polygon must clearly indicate the shape of the curve so that the designer can interactively control the curve by slightly moving the control points. For this specific purpose, we propose the condition number of the corresponding matrix as an estimating criterion.