Computer Aided Engineering Design

142 COMPUTER AIDED ENGINEERING DESIGN

Note that f(t) = f′(t) = f′′(t) =... = fm–2(t) = 0 at t = 0. However, fm–1(t = 0+) = (m–1)! while

fm–1(t = 0–) = 0 implying that fm–1(t) is discontinuous at t = 0. Thus, by definition, f(t) = tm+–1 is a
spline of order m over the entire range of t. Next, for some knot tj, consider the function f(tj;t) =

(tj–t)+m–1 which is continuous at t = tj and so are its derivatives up to m– 2 as above. Thus, (tj–t)+m–1
is a spline of order mas well (Figure 5.8(b)). A linear combination of such splines considered over
a knot span t 0 ,... , tn, that is

ψα() = tttr=0 ( – )+–1

n

rr

Σ m (5.19)

with non-zero constants αr will be a spline of order m. This was first established by Sohenberg and
Whitney in 1953. A B-spline basis function can be computed as the mth divided difference of the
truncated power function f(tj;t) = (tj–t)+m–1. Considering t as constant and computing the m th
divided difference for tj = ti–m, ti–m+1,... , ti, we have

ft t t t

tt

wt

imim i t

r

m

irm m

irm

[ , ,... , ; ] =

(– )

()

• –+1 = )
  =0

+– +–1

+–

Σ

′

ψ( (5.20)

wherew(t) = (t–ti–m)(t–ti–m+1)... (t–ti) and w′(t) = dw/dt. That ψ(t) is a linear combination of
individual splines of order mand thus ψ(t) by itself is a spline of the same order is established by
Eq. (5.19). Further, ψ(t) is a B-spline basis function Mm,i(t) for the following reasons:

(i) ψ(t) = 0, for t > ti since the individual truncated functions (ti+r–m–t)+m–1,r = 0,... , m are all
zero.
(ii) ψ(t) = 0, for t≤ti–m since ψ(t) is the m th divided difference of a pure (m–1) degree polynomial
int. The mth divided difference is representative of (but not equal to) the mth derivative which
is zero for a pure polynomial of degree upto m– 1.
(iii) We can further show that ψ(t) is standardized, or

t

t

t

t

imim i

im

i

im

i

tdt ft t t tdt

–– m

( ) = [ – , –+1,... , ; ] =^1

∫∫

ψ (5.21)

For this, Peano’s theorem for divided differences may be used.

( – 1)! [ – , –+1,... , ; ] = ( ) ( )

• 
  

mgttimim tti tgtdt

t

t

m

im

i

∫

ψ (5.22)

for any g(t). Choosing g(t) = tm,gm(t) = (m)!. Further, the mth divided difference of tm is 1 (which
can be verified using hand calculations for smaller values of m). Thus, Eq. (5.22) becomes

( – 1)! = ( )! ( ) ( ) =^1

––

mm tdt tdt

t m

t

t

t

im

i

im

i

∫∫

ψψ⇒

Example 5.3.Show, using plots, that ψ(t) = f[t 0 ,t 1 ,t 2 , t 3 ;t] = Σ

r

r

r

tt

=0 wt

3

+

(– )^2

′()

is a quadratic B-spline

basis function with w(t) = (t–t 0 )(t–t 1 )... (t–t 3 ) and w′(t) = dw/dt. Assume that t 0 < t 1 < t 2 < t 3.