Non-Uniform Rational B-Spline)

The B-spline method cannot accurately describe quadratic curves and spherical surfaces, but rational B-spline can solve this problem. The uniform and non-uniform B-spline described above is a subset of the non-uniform rational B-spline discussed in this section. Versperille (1975) at the University of sirachouz in the United States was the earliest study of rational B-spline method, while versperille (1983), who pushed the approach to practical use, mainly focused on the in-depth research by piegl and tieller (1989.
The abbreviation of the non-even rational B-spline (non-unim M rational B-spline) method, the main contribution of the method is to unify the B-spline method used to describe the Free Curve and the mathematical method used to accurately represent the quadratic curve and the quadratic surface. In view of the powerful functions and potential of the Shape Definition in the form field. In the geometric definition STEP standard of industrial products officially promulgated by the International Standards Organization (ISO) in 1991, the only Representation Method for a Free Curve or surface is to be defined as a place where. In 1983, tieller extended the B-spline to the four-dimensional homogeneous coordinate space to generate a rational B-spline curve. In the four-dimensional space

All the control points in the same space. If Phi = (hxi, BYI, hzi, H), the curves in the three-dimensional space are shown below:

The curve depicted above is called a rational B-spline curve.
Figure 2.13 shows the diagram of rational base function RI and K (u) in a range. The hi in the above formula can also be seen as the Weight of item L. Figure 2.14 shows the geometric push-pull effect produced by changing the weight h in a rational Cubic B spline.
If the nodes in the preceding formula are uniformly distributed with the basic functions Ni and K (U), B (U) is called the even rational B-spline (urb ). If it is not uniform, it is called (nurb) Non-Uniform Rational B-spline. The uniform distribution of the basis function, that is, the uniform selection of the node vector on the parameter axis, makes the generation curve have some limitations (such as the curve length corresponding to the node interval ), the uneven distribution of base function parameters can change this situation. We can choose to make the corresponding curve segment equal to or near the same length, and give better control from B (Figure 2.15)

The non-uniform rational B-spline has the following four characteristics:
(1) All the advantages of B-Spline are retained in the non-uniform rational B-spline.
(2) perspective immutability. The curve or surface generated after the control point passes the perspective transformation is equivalent to the re-transformation of the original curve or surface.
(3) precise representation of Quadratic Surfaces such as the sphere. Other B-Spline methods can only represent the shape of the sphere, while the NURB can not only represent the Free Curve and surface, but also accurately represent the shape of the sphere.
(4) more shape control degrees of freedom. NURB gives more degrees of freedom to control shapes and can be used to generate various shapes.