Computer Aided Engineering Design

148 COMPUTER AIDED ENGINEERING DESIGN

(E) Partition of unity*: The sum of all non-zero order p B-spline functions over the span
[ti,ti+1) is 1

Example 5.5.Demonstrate the partition of unity property for quadratic normalized B-splines.
We know from above that over [ti,ti+1), the quadratic normalized B-splines N3,i+ 1 (t),N3,i+ 2 (t) and
N3,i+ 3 (t) are non-zero.

(D) On any span [ti,ti+1), at most p order p normalized B-spline functions are non-zero
This follows from the local support property mentioned above. For any r,Np,r(t)≥ 0 in the knot span
[tr–p, tr). So that [ti,ti+1) is contained in [tr–p, tr), it must be ensured that there is at least one order p
B-spline with ti as the first knot and at least one with ti+1 as the last knot. Thus, r–p = i and r = i+1
provide the range in r, that is, r = i+1,... , i+p for which Np,r(t) is non-zero in [ti,ti+1). This adds
up to p B-splines. Figure 5.11 demonstrates this property for p = 4 (a normalized cubic B-spline). It
is this property that provides local control when reshaping B-spline curves discussed later.

ti–4 ti–3 ti–2 ti–1 ti ti+1 ti+2 ti+3 ti+4 t

N4,i+4(t)

N4,i+2(t) N4,i+3(t)

N4,i+1(t)

Figure 5.11 Schematic of the normalized fourth order B-splines that are non-zero over [ti,ti+1)

ti–2 ti–1 ti ti+1 ti+2 ti+3 t

N3,i+3(t)

N3,i+1(t) N3,i+2(t)

Figure 5.12 Schematic of the normalized third order non-zero B-splines over [ti,ti+ 1)

Using Eq. (5.28), first N3,i(t) is computed.

Nt

tt

tt

tt

i tt

i

ii

i

i

ii

2, –1 i

–3

–2 –3

–2

–1

–1 – 2

() = –1

• 
  
  
  
  • 
    
    
    
    • 
      
    
    
    
    
    
    
    • δδ–
    
    
    
    
  
  
  
  

Nt

tt

tt

tt

i tt

i

ii

i

i

i i

2, i

–2

–1 – 2

–1

–1

() =

• 
  
  
  
  • 
    
    
    
    • 
      
      
      
      • 
        
      
      
      
      
    
    
    
    
  
  
  
  


• 
  

  δδ

  
  

Nt

tt

tt

Nt

tt

tt

i i Nt

ii

i

i

i i

3, i

–3

–1 –3

2, –1

–2

() = 2,

• 
  
  
  
  • () +
    
    
    • 
      
    
    
    
    
  
  
  
  


• 
  

  ()

  
  

*This property can be proved using mathematical induction for a generic case.