Computer Aided Engineering Design

104 COMPUTER AIDED ENGINEERING DESIGN

(b) Partition of Unity and Barycentric Coordinates: Irrespective of the values of u, the Bernstein
polynomials sum to unity, that is

Σ

i

n

i

Bun

=0

() = 1 (4.35)

From Binomial expansion

[(1 – ) + ] = (1 – ) + (1 – ) +... +

!

!( – )!

uu u n u u–1 (1 – )– +... +

n

rn r

nn n uunrrnu

1 = nC 0 (1 – u)nu^0 + nC 1 (1 – u)n–1u + nC 2 (1 – u)n–2u^2 + nCn (1 – u)^0 un

or 1 = Bu Bu 01 nn( ) + ( ) +... + Bunn( )

Eqs. (4.33)-(4.35) suggest that the point b 0 n on the nth degree Bézier curve is the weighted linear
combination of the n + 1 data points b 0 ,b 1 ,... , bn with respective weights as Buin(), where Buin()
are all non-negative and sum to unity. These weights are analogous to the point masses placed,
respectively, at b 0 ,b 1 ,... , bn whose center of mass is located at b 0 n. For this reason, such weights
are known as barycentric coordinates, the term barycenter implying the center of gravity. Note that
as the center of mass always lies within the convex hull of the locations of individual point masses,

so does b 0 n lie in the convex hull of data points for values of u between 0 and 1. The convex hull of

a set of points is the smallest convex set that contains all given points. Any line segment joining two
arbitrary points in a convex set also lies in that set.

Figure 4.14 Plot of Bernstein polynomials Buin() for: (a) n = 3 and (b) n= 4

Bu 03 ()

Bu 13 () Bu 23 ()

Bu 33 ()

Bu 04 ()

Bu 14 ()

Bu 24 () Bu^3

(^4) ()
Bu 44 ()
0 0.2 0.4 0.6 0.8 1
u
(b)
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
u
(a)
Since 0 ≤ 1 – u≤ 1 as well,
Bu Cu u
n
i in iuu
nn( ) = i ini(1 – ) =! ini
!( – )! (1 – )^0
––≥
Non-negativity can be appreciated by the plots of Buiii^34 ( ), = 0,... , 3 and Bui( ), = 0,... , 4 in
Figure 4.14.