Computer Aided Engineering Design

SPLINES 145

t

t

ii

t

t

i i i

i

i

i

i

MtdtMt dtMtt t

–1 –1

∫∫1, () = 1, () = 1,()( – –1) = 1

or Mti tt
i i
1,
–1

() =^1

• for t∈ [ti–1,ti); (5.25)

= 0 elsewhere
Combining Eqs. (5.24) and (5.25), the recursion relation for a B-spline basis function may be
written as

Mti tt

i i

1,

–1

() = –^1 for t∈ [ti–1,ti);

= 0 elsewhere

Mk,i(t) =

tt

tt

Mt

tt

tt

ik Mt

i ik

ki

i

i ik

ki

• 
  
  
  
  • () +
    
    
    • 
      
    
    
    
    
  
  
  
  


• 
  


• ()


• 
  

  –1, –1

  
  


• 
  

  –1, for t∈ [ti−k,ti); (5.26)

  
  

= 0 elsewhere

5.6.1 Normalized B-Spline Basis Functions
The normalized B-spline weight Nk,i(t), which are used more frequently in the design of B-spline
curves may be computed as

Nk,i(t) = (ti–ti–k)Mk,i(t) (5.27)

From Eqs. (5.25) and (5.27), N1,i(t) = (ti–ti–1)M1,i(t) = 1 for t in the range [ti–1,ti) and N1,i(t) = 0 for
all other values of t. We may combine the two results as N1,i(t) = δi, where δi = 1 for t∈ [ti–1,ti) and
δi = 0 elsewhere. For higher order normalized B-splines, the recursion relation can be derived using
Eqs. (5.24) and (5.27). Starting with (5.24)

Mt

tt

tt

Mt

tt

tt

ki ik Mt

i ik

ki

i

i ik

, ki

• 
  
  
  
  • –1, –1
    
    
    • 
      
    
    
    
    
  
  
  
  

() = –1,

• 
  
  
  
  • () +
    
    
    • 
      
    
    
    
    
  
  
  
  


• 
  

  ()

  
  

Table 5.2 Recursion to compute B-spline basis functions

[ti–k,ti–k+1) M1,i–k+1(t)
M2,i–k+2(t)
[ti–k+1,ti–k+2) M1,i–k+2(t)
M2,i-k+3(t)
[ti–k+2,ti–k+3) M1,i–k+3(t)
Mk–1,i–1(t)
M... Mk,i(t)
Mk–1,i(t)
[ti–3,ti–2) M1,i–2(t)
M2,i–1(t)
[ti–2,ti–1) M1,i–1(t)
M2,i(t)
[ti–1,ti) M1,i(t)