1.4 B-spline curves and surfaces

The Bézier representation has two main disadvantages. First, the number of control points is directly related to the degree. Therefore, to increase the complexity of the shape of the curve by adding control points requires increasing the degree of the curve or satisfying the continuity conditions between consecutive segments of a composite curve. Second, changing any control point affects the entire curve or surface, making design of specific sections very difficult. These disadvantages are remedied with the introduction of the B-spline (basis-spline) representation.

Early fundamental work on the B-spline basis functions was performed almost 50 years ago by Schoenberg [368], and this was followed by development of fundamental algorithms by Cox [67] and de Boor [72,73]. B-splines in the context of Computer Aided Geometric Design were proven to be a viable and attractive representation method by many pioneers of this field, such as Riesenfeld [345,130], Boehm [33], Schumaker [369] and many subsequent researchers.

In this section, we provide definitions and the basic properties and algorithms of B-splines. However, we do not deal with fitting, approximation and fairing methods using B-splines which are very important in their own right. For these topics, there are specialized books, monographs and proceedings and a large variety of papers [365,175,92,314,45].