Constructive implicit fitting

In this paper, we present a constructive method for fitting both an explicit and an implicit curve or surface to a set of scattered points by using gate functions. With this technique, the data are first partitioned with geometric primitives into small data sets such that each sub-data set can be well fitted by a simple algebraic or non-algebraic shape. These simple shapes are then blended to form an overall fitting for the given data. Compared with some conventional fitting techniques, the proposed method has the following distinct features. First of all, a preset accuracy can always be achieved if the data set is sufficiently finely partitioned. Secondly, a large data set can be dealt with as several small data sets, which can significantly reduce the complexity of the shape to be fitted. Thirdly, the proposed fitting technique can also be used to increase the fitting speed by using simple fitting techniques for each sub-data set. Fourthly, the degree of smoothness of the fitted function can be adjusted to be as smooth as one wishes. Furthermore, the fitting technique provides direct support for parallel computation in curve and surface fitting. When a large data set is partitioned into smaller data sets, these small data sets can then be fitted simultaneously over a parallel system. This will greatly reduce computation time.