Computer Aided Engineering Design

114 COMPUTER AIDED ENGINEERING DESIGN

As an example, the control points for a cubic Bézier segment in 0 ≤u≤c are determined. From
Eq. (4.50)

k = 0: DP 00 = 00 ⇒p 0 = b 0

k = 1: cDP^10 = 01 ⇒c(b 1 – b 0 ) = (p 1 – p 0 )

⇒p 1 = (1 – c)b 0 + cb 1

k = 2: c^2 DP^20 = 02 ⇒c^2 (b 2 – 2b 1 + b 0 ) = (p 2 – 2p 1 + p 0 )

⇒p 2 = (1 – c)^2 b 0 + 2c(1 – c)b 1 + c^2 b 2

k = 3: c^2 DP^30 = 03 ⇒c^3 (b 3 – 3b 2 + 3b 1 – b 0 ) = (p 3 – 3p 2 + 3p 1 – p 0 )

⇒p 3 = (1 – c)^3 b 0 + 3c(1 – c)^2 b 1 + 3c^2 (1 – c)b 2 + c^3 b 3 (4.51)

Comparing Eq. (4.51) with (4.32), it can be observed that pbpbpb 0 = 00 , = , = 1 10 2 02 and pb 3 = 03
foru = c. In general, pbk = k 0 , = 0,... , .kn Geometrically, therefore, the new control points

p 0 ,p 1 ,... , pn for the first Bézier segment are the intermediate de Casteljau points, bb 00 , 01 ,... , b 0 n
foru = c which in Figure 4.13, in the triangular schema to compute the de Casteljau points, correspond
to the top edge of the triangle.
For control points q 0 ,q 1 ,... , qn of the second segment, the Bézier curve for c≤u≤ 1 may be re-
parameterized with u′ such that u= 1 – (1 – c) (1 – u). Thus, when u′ = c,u= 0 and for u′= 1, u =

1. The Bézier segment r 2 (u) for c≤u≤ 1 is

r 2

=0 =0

( ) = uB(1 – (1 – )(1 – ))cu = Bu( )

i

n

i

n

i i

n

i

n

ΣΣbqi (4.52)

Identical to the treatment for the first segment, here, the derivatives of the two curves can be matched
atu′ = 1. Consider the kth derivative as per Eq. (4.45) and after implementing the chain rule

ddu nnkk nki Bu c

nk

i

nk

i

r kk

(^2) =0

• / ′ = ⎡( – 1)... ( – + 1) – ( ) (1 – )
  ⎣

⎢

⎤

⎦

Σ D ⎥

= 2 / = ( – 1)... ( – + 1) ( )

=0

• ddunnkk nk Bu–
  i

nk

i

nk

i

rQΣ ′ k (4.53a)

whereQij are the differences in qi related in a manner similar in Eqs. (4.44d) and (4.49) as

QQij = Qij+1–1 – ij–1, = 1,... , ; = 0,... , – , with jninjQq^0 i = i (4.53b)

Atu′ = 1, Eq. (4.53a) becomes

(1 – )cknkDQnkk–– = nkk , = 0,... , (4.54)

To illustrate the computations, the control points for a cubic Bézier segment in c≤u≤ 1 can be
determined as