### Basis Functions of B-spline Curvesfor p = 0,··· ,3 with 0,···, 3 Middle Knots

### Definition

A pth-degree B-spline curve is defined by

where *P** _{i}* are the control points, and

*N*are the

_{i,p}(u)*p*th-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

*i*th B-spline basis function of

*p*th-degree, denoted by

*N*, is defined as

_{i,p}(u)*N _{i,0}(u) *is a step function which only equal to 1 at its own interval [

*u*,

_{i}*u*). And

_{i+1}*N*at higher degree is a linear combination of two (

_{i,p}(u)*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*are given in the

_{i,p}(u)**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