Computer Aided Engineering Design

SPLINES 147

thus, N2,i(t)≥ 0 for t in [ti–k,ti) (5.30)

Likewise, N2,i–1(t)≥ 0 for t in [ti–k,ti) (5.31)

Next, assume that Eqs. (5.30) and (5.31) are true for the (k–1)th order normalized splines, that is

Nk–1,i(t)≥ 0, t∈ [ti–k,ti) and Nk–1,i–1(t)≥ 0, t∈ [ti–k,ti) (5.32)

Then, using Eq. (5.28)

Nt

tt

ttNt

tt

ki tt Nt

ik

i ik ki

i

, i ik ki

• –1 – –1, –1 – +1 –1,

() =

• 
  
  
  
  • 
    

    () +

    
    
  
  
  
  


• ()

From Eq. (5.32) and additionally, since

tt

tt

ik

i ik

• 
  
  
  
  • 
    
    
    
    • –1 –
    
    
    
    
  
  
  
  

≥ and

tt

tt

i

i ik

• 
  


• 
  

  0

  
  


• +1

≥ for t∈ [ti–k,ti),

Nk,i(t)≥ 0 for t in [ti–k,ti).

Example 5.4.Verify using plots, the non-negativity property of N4,i(t) with knots ti–4 = 0, ti–3 = 1,
ti–2 = 2, ti–1 = 3, ti = 4.
The plot for various normalized B-splines is shown in Figure 5.10. N1,j(t),j = 1,... , 4 are step
functions which are equal to 1 in [tj–1,tj) and are zero otherwise. N2,j(t),j = 2,... , 4 are linear
triangle-shaped functions, N3,j(t),j = 3, 4 are the inverted bell-shaped quadratics while N4,4(t) is the
cubic B-spline function (thickest solid line). Note that all splines are non-zero in their domains of
definition.

012 3 4

t

2

1.5

1

0.5

0

N3, 3(t) N3, 4(t)

N4, 4(t)

N2, 2(t) N2, 3(t) N2, 4(t)

N1, 1(t) N1, 2(t) N1, 3(t) N1, 4(t)

Figure 5.10 Plot of the normalized B-splines constituting N4,i(t) for uniform knot spacing

(C) Local support: Nk,i(t) is a non-zero polynomial in (ti−−−−−k, ti)
From Eq. (5.20) Mk, i(t) = 0 for t≥ti. Since Mk,i(t) is the kth divided difference in [ti–k,ti) of a linear
combination of pure polynomials of degree k–1 each,Mk,i(t) = 0 for t≤ti−k. Thus, from Eq. (5.27),
Nk,i(t) = 0 for t≤ ti–k and t≥ti. From Eq. (5.21), the integral of Mk,i(t) over the interval is 1/k. This
implies that Nk,i(t) over [ti–k,ti) is at least not zero entirely in the interval. Additionally, the non-
negativity property above suggests that Nk, i(t) does not have any root in [ti–k, ti) and so is a non-zero
polynomial in (ti–k, ti). Thus, in a given parent knot sequence (t 0 ,t 1 ,... , tn), all B-spline functions
Nk, i(t),i = k,... , n (k≤n) have their subdomains in [ti–k, ti) wherein they are non-zero.