Computer Aided Engineering Design

DESIGN OF CURVES 103

The computation of intermediate de Casteljau’s points for degree n Bézier curve can be illustrated by

the triangular scheme in Figure 4.13. de Casteljau arrived at this result in 1959 at Citroen (a French

car company), where he was working on the shape design of curves. He, however, never published

his procedure until the internal reports were discovered in 1975. From Eqs. (4.31) and (4.32), by

inspectionb 0 n can be written as

bbb 0 n = i=0 (1 – )– = =0 ( )

n

n

i

ni i

i i

n

i

n

ΣΣCuu Bui (4.33)

where Buinn( ) = Ci(1 – )u uni i– are termed as Bernstein polynomials. Eq. (4.33) defines Bézier

curves (discussed in Section 4.4) that Bézier proposed independently using Bernstein polynomials.

In de Casteljau’s work, no assumption about the Bernstein type blending functions was made, yet

repeated linear subdivisions of the control polylines resulted in the same conclusion as Bézier’s work.

Figure 4.13 Triangular scheme for (a) n = 2, (b) n = 3 and (c) to compute intermediate

de Casteljau points. Schema on top right is representative of the weighted

linear combination of de Casteljau’s points

(a)

(b)

b 00

b 01

b b^02

1

0

b 11

b 20

1 – u

1 – u

1 – u u

u

u

b 00

b 10

b 20

b 30

b 03

b 01

b 02

b 11

1 – u

u

b 21

1 – u

1 – u

1 – u

1 – u

1 – u

u

u

u

u

u

4.2.2 Properties of Bernstein Polynomials

Bernstein polynomials play a significant role in predicting the segment’s shape and by understanding

their behavior, a great deal can be learnt about the Bézier curves.

(a) Non-negativity: For 0 ≤u≤ 1, Buin() are all non-negative. (4.34)

c 3 = (1 – u)c 1 + uc 2

c 1

c 2 u

1 – u

b 0 n

b 1 n–1

b 0 n–1

bn^2 –2

bn^1 –2

bn^1 –1

bn^0 –2

bn^0 –1

bn^0

b 02

b 11

b 01

b 00

b 10

b 20

(c)