Computer Aided Engineering Design

138 COMPUTER AIDED ENGINEERING DESIGN

Mm,i(t)

ti–4 ti–3 ti–2 ti–1 ti t

Figure 5.6 Schematic of a B-spline basis function of order 4

to the Bernstein polynomials, and thus can be used as weighting or basisfunctions. For this reason,
standardized splines are also termed as basis-orB-splines. In general, a B-spline of order mwith the
last knot as ti can be denoted as Mm,i(t). Similar to a cubic B-spline, Mm,i(t) may be computed with
the end conditions, Mm,i(t) = dMm,i(t)/dt = d^2 Mm, i(t)/dt^2 =... = dm–2Mm, i(t)/dtm–2 = 0 at both ends,
i.e., 2(m–1) conditions with continuity conditions Mm,i(t),dMm,i(t)/dt,d^2 Mm,i(t)/dt^2 ,... , dm–2Mm,i(t)/
dtm–2 continuous at interior knots. As mentioned above, this method of computing B-splines is quite
tedious and requires many algebraic manipulations. Alternatively, the divided difference approach
may be employed.

5.5 Newton’s Divided Difference Method

The divided difference scheme (discussed briefly in Section 3.1) uses the following curve interpolation
approach for given points (xi,yi),i = 0,... , n– 1. A polynomial of degree n– 1 can be written as

y = pn– 1 (x)

=α 0 + α 1 (x– x 0 )+α 2 (x – x 0 ) (x– x 1 ) +... + αn–1(x– x 0 ) (x– x 1 )... (x– xn–2) (5.13)

where the unknown coefficients α 0 ,α 1 ,... , αn–1 can be determined using the following substitutions.

y 0 =pn–1(x 0 ) = α 0

y 1 =pn–1(x 1 ) = α 0 + α 1 (x 1 – x 0 )⇒α 1 =

yy

xx

10

10

01234

t

0.25

0.2

0.15

0.1

0.05

0

Φ(t)

Figure 5.5 Computed normalized cubic spline with knots ti = i,i = 0,... ,4 for Example 5.2

Φ 0 (t)

Φ 1 (t)

Φ 2 (t)

Φ 3 (t)