B-spline Curve Fitting: Application to Cartographic Generalization of Maritime Lines

Generalization is the process of abstraction applied when the scale of a map is changed. It involves modifications of data in such a way that the data can be represented in a smaller space, while best pre- serving geometric as well as descriptive characteristics. A map is an abstracted model representing the geometric reality. The smaller the scale, the more schematic the representation. Line cartographic generalization deals with graphic representation of lines. Many al- gorithms are available for an automated line cartographic general- ization. Instructions for using these algorithms are often complex and representations applied ill-adapted to some generalization pro- cesses. In this paper, we explain the advantages of using B-spline curves in a line generalization process. We focus on processing of line cartographic generalization operators in a maritime context. Cartographic generalization includes the whole processings en- countered when the scale of the map is changed into a smaller scale. We should produce a legible map which is as close as pos- sible from reality. The tools currently available for automated car- tographic generalization resemble those of the manual generaliza- tion. A catalogue of cartographic generalization operators has been proposed (23), including selection/elimination, aggregation, struc- turing, compression (or filtering), smoothing, exaggeration, carica- turing, enlargement and displacement. One can essentially distin- guish between two approaches for the implementation of the work- ing tools in generalization. One is automatic while the other is in- teractive. The generalization automation has been studied for over twenty years. The difficulties of providing an automatic solution points out the complexity of the problem. The second section of the paper deals with the representations used for data modelling. Subsection 2.1 is devoted to the represen- tation by means of a list of points. Most generalization algorithms have been developed focusing on the manipulation of vectors. Rep- resentation by means of a list of points does not provide fair mod- elling of curves which may have complex and varying shapes. In addition, this representation is often ill-adapted to some generaliza- tion process. In subsection 2.2, we suggest a different representation based on B-spline curves. The third section of the paper deals with the application of B- spline representation in processing of line cartographic generaliza- tion operators in a maritime context. In subsection 3.1, we focus our attention on data compression using a bisection method on the number of control points. Line smoothing and displacement opera- tors are developed in subsection 3.2. The strategy is based on a me- chanical approach. The curve displacement is obtained through the displacement of control points. Internal and external forces are ap- plied at control points in order to produce the desired deformation. Lastly, we introduce a technique for curve aggregation (subsection 3.3).