We present an efficient method to automatically compute a smooth approximation of large functional scattered data sets given over arbitrarily shaped planar domains. Our approach is based on the construction of a ¢ ¤ £ -continuous bivariate cubic spline and our method offers optimal approximation order. Both local variation and nonuniform distribution of the data are taken into account by using local polynomial least squares approximations of varying degree. Since we only need to solve small

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... ar systems and no triangulation of the scattered data points is required, the overall complexity of the algorithm is linear in the total number of points. Numerical examples dealing with several real world scattered data sets with up to millions of points demonstrate the efficiency of our method. The resulting spline surface is of high visual quality and can be efficiently evaluated for rendering and modeling. In our implementation we achieve real-time frame rates for typical fly-through sequences and interactive frame rates for recomputing and rendering a locally modified spline surface. Simplicity: The method should be easy to implement. Although many approaches have been developed in the last 30 years, the literature shows that it is a difficult task to meet all of the above goals by using one single method. In fact, the algorithms proposed in the literature typically have at least one of the following drawbacks: limitations in approximation quality, severe restrictions on the number of points, limited visual quality of the resulting surface, high computation times, and restrictions on the domain and distribution of the data. In this paper, we develop a new approach to scattered data fitting which is based on differentiable bivariate cubic splines. We decided to construct a smooth surface since such surfaces look more pleasant and have nice properties for further processing and rendering. The method we propose belongs to the class of so-called two-stage methods [40]: In the first step of the algorithm, we compute discrete least squares polynomial pieces for local parts of the spline 9 by using only a small number of nearby points. Then, in the second step, the remaining polynomial pieces of