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)