Computer Aided Engineering Design

SPLINES 143

It is required to show that (a) ψ(t) is a quadratic spline, (b) ψ(t) = 0 for t < t 0 and t > t 3 , (c) ψ(t)

is non-negative for all t and, finally, (d)
t

t

tdt

0

3

() =^1

∫ 3

ψ.

(a) It is known that the truncated power functions of order m (degree m– 1) are continuous up to the
m– 2 derivatives so that f[t 0 ,t 1 ,t 2 , t 3 ;t] is a quadratic spline.
(b) Further, to show that it is a B-spline, we consider the expanded form of f[t 0 ,t 1 ,t 2 , t 3 ;t], that is,

ψ( ) = [ , , , ; ] =

(– )

()

+

(– )

()

+

(– )

()

+

(– )

(^0123) ()
0 +^2
0
1 +^2
1
2 +^2
2
3 +^2
3
tfttttt
tt
wt
tt
wt
tt
wt
tt
′′′′wt
Fort > t 3 , all truncated functions in ψ(t) are zero and thus ψ(t) is zero. For t< t 0 , all truncated
functions are pure quadratic functions. Ignoring the truncation (+) sign
ψ( < ) =
(– )
()

• (– )
  ()


• 
  

  (– )
  ()

  
  


• 
  

  (– )
  (^0) ()
  0
  2
  0
  1
  2
  1
  2
  2
  2
  3
  2
  3
  tt
  tt
  wt
  tt
  wt
  tt
  wt
  tt
  ′′′′wt
  Noting that the above is the third divided difference of a quadratic polynomial (tj–t)^2 , the tabular
  form may be used to compute ψ(t<t 0 ).
  tvalues f[tj;t] 1st differences 2nd differences 3rd differences
  t 0 (t 0 – t)^2
  [(t 1 – t)^2 – (t 0 – t)^2 ](t 1 – t 0 )
  =t 0 + t 1 – 2t
  t 1 (t 1 – t)^2 (t 2 – t 0 )(t 2 – t 0 ) = 1
  [(t 2 – t)^2 – (t 1 – t)^2 ](t 2 – t 1 )
  = t 1 + t 2 – 2t 0 = ψ(t <t 0 )
  t 2 (t 2 – t)^2 (t 3 – t 1 )(t 3 – t 1 ) = 1
  [(t 3 – t)^2 – (t 2 – t)^2 ](t 3 – t 2 )
  = t 2 + t 3 – 2t
  t 3 (t 3 – t)^2
  Thus,ψ(t<t 0 ) = 0 which is expected by definition (Eq. (5.20)).
  (c) Showing that ψ (t) is non-negative is deferred until the next section though we may be convinced
  by referring to the plot in Figure 5.9 for t 0 = 0, t 1 = 2, t 2 = 3 and t 3 = 6.
  (d) Showing that ψ (t) is standardized can be done using Eq. (5.22) for m = 3.

  
  

5.6 Recursion Relation to Compute B-Spline Basis Functions

Eq. (5.20) provides two ways to compute the B-spline basis function of order m either by computing
the divided differences in the tabular form as in the left hand side or computing it algebraically as in
the right hand side. The third alternative proposed by Cox and de Boor (1972) is the recursion
relation that can be derived from divided differences. Using Leibnitz result on divided differences of
the product of two functions h(t) = f(t)g(t), we have

h[t 0 ,t 1 ,... , tk] = rΣ

k

=0f[t^0 ,t^1 ,... , tr]g[tr,tr+1,... , tk]