Computer Aided Engineering Design

146 COMPUTER AIDED ENGINEERING DESIGN

⇒

Nt

tt

tt

tt

Nt

tt

tt

tt

Nt

tt

ki

i ik

ik

i ik

ki

i ik

i

i ik

ki

i ik

,

• 
  
  
  
  • 
    
    
    
    • 
      
    
    
    
    
  
  
  
  

–1, –1

–1 ––

–1,

–+1

()

• 
  

  =

  
  

   -

  
  

()

• 
  
  
  
  • 
    
    
    
    • 
      
    
    
    
    
  
  
  
  


• 
  

()

• 
  

⇒ Nt

tt

ttNt

tt

ki tt Nt

ik

i ik ki

i

, i ik ki

• –1 – –1, –1 –+1 –1,

() =

• 
  
  
  
  • 
    

    () +

    
    
  
  
  
  


• ()

The recursion relation to compute normalized B-splines is then

N1,i(t) = δi such that δi = 1 for t∈ [ti− 1 ,ti)

= 0, elsewhere

Nt

tt

tt

Nt

tt

tt

ki ik Nt

i ik

ki

i

i ik

, ki

• –1 –
  –1, –1
  –+1

() = –1,

• 
  
  
  
  • () +
    
    
    • 
      
    
    
    
    
  
  
  
  


• 
  

  () (5.28)

  
  

Normalized B-splines may be used as basis functions to generate B-spline curves as Hermite and
Bernstein polynomials are used in designing Ferguson and Bézier curves, respectively (Chapter 4).
In that regard, a study of the properties of B-spline basis functions becomes essential. It may be
mentioned that in some publications the notation Ni,k(t) is used instead of Nk,i(t), where k is the
degree of the polynomial in t and ti is the first knot value.

5.7 Properties of Normalized B-Spline Basis Functions

(A)Nk, i(t)is a degree k−−−−−1 polynomial in t
From Eqs. (5.20) and (5.24), Mk,i(t) is a piecewise polynomial of degreek–1 in the knot span
[ti–k,... ti) and therefore from Eq. (5.27), Nk,i(t) is a polynomial of degreek–1.

(B) Non-negativity: For all i,k and t,Nk, i(t) is non-negative
The property can be deduced by induction. In a given knot span ti–k < ti–k+1 <... < ti,

N1, i(t) = 1 for t∈ [ti–1,ti)

N1, i(t) = 0 elsewhere from Eq. (5.28)

Thus,N1,i(t)≥ 0 in [ti–k,ti)

Similarly, N1,i–1(t) = 1 for t∈ [ti–2,ti–1) and N1,i–1(t) = 0 elsewhere

Also,N1,i–2(t) = 1 for t∈ [ti–3,ti–2) and N1,i–2(t) = 0 elsewhere

Thus, both N1,i–1(t)≥ 0 and N1,i–2(t)≥ 0 in [ti–k, ti) (5.29)

From Eqs. (5.28) and (5.29),

Nt

tt

ttNt

tt

i ttNt

i

iii

i

2, i i i

–2

–1 – 2 1, –1 –1 1,

() =

• 
  
  
  
  • 
    

    () +

    
    
  
  
  
  


• () for t∈ [ti–2,ti) and N2,i(t) = 0 elsewhere.

Now, Nt
tt
i tt
i
ii
2,
–2
–1 – 2

() =

• 
  
  
  
  • 0 ≥ for t∈ [ti–2,ti–1]
  
  
  
  

=

• 
  


• 
  

  0
  –1

  
  

tt

tt

i

i i

≥ for t∈ [ti–1,ti)

= 0 elsewhere