Bézier Approximate Merging by Interval Curves

This paper mainly deals with the approximate merging problem of multiple adjacent Bézier curves with different degrees by new perspective—a single interval Bézier curve, which is a frequently seen problem in modeling, geometric approximation and data transformation. At first, two methods are given in this paper as theoretical models. One gives the result of the interval Bézier curve by one-sided approximation, which can directly get the values of Bézier control points. The other uses the unified matrix representation for precise merging. The approximate merging center curve is further derived based on matrix operation and the error curve is given by both constant and unconstant interval. Then, several examples are provided to demonstrate the algorithms, which show that the methods in this paper both achieve satisfying merging results and have a good prospect in geometric approximation and data transformation. These methods will be chosen for different results and reasons. More and more, it can be proved that these methods can also be used in the merging process of multiple adjacent rational Bézier curves, in the merging process of multiple adjacent 3D Bézier curves, in the approximate merging problem by a disk Bézier curve and in the approximate merging problem of multiple adjacent Bézier surfaces.