Definition

            A pth-degree B-spline curve is defined by

where P_i are the control points, and N_i,p(u) are the pth-degree B-spline functions defined on the knot vector (m+1 knots):

U is a nondecreasing sequence of real numbers. The u_i are called knots, and U is called the knot vector. The ith B-spline basis function of pth-degree, denoted by N_i,p(u), is defined as

N_i,0(u) is a step function which only equal to 1 at its own interval [u_i, u_i+1). And N_i,p(u) at higher degree is a linear combination of two (p−1)th-degree basis functions. The combination of step functions leads to their local support property. It should be noted that 0/0 is specially defined to be equal to 0 in the computation.

Analytic Expression

The analytic expressions of N_i,p(u) when p=0, 1, 2, 3 are listed in the Table 1, 2, 3, and 4, respectively. The middle knots means that their knot value are neither 0 nor 1. Here the knot vector is defined with uniform intervals. And the graphs of N_i,p(u) are given in the Figure 1, 2, 3, and 4, respectively. In the graphs, the dashed vertical lines separate the different knot intervals. When we add the expressions in the same interval and same degree together, the summation will always be one, which is called one of their properties, partition of unity. According to the graphs of the basis functions, their important properties are illustrated, such as, local support property and nonnegativity.

Table 1: Analytic Expression of N_i,p(u) for 0 middle knot, Bézier function

Figure 1: Graph of N_i,p(u) for 0 middle knot

Table 2: Analytic Expression of N_i,p(u) for 1 middle knot

Figure 2: Graph of N_i,p(u) for 1 middle knot

Table 3: Analytic Expression of N_i,p(u) for 2 middle knots

Figure 3: Graph of N_i,p(u) for 2 middle knots

Table 4: Analytic Expression of N_i,p(u) for 3 middle knots

Figure 4: Graph of N_i,p(u) for 3 midd knots