Computer Aided Engineering Design

110 COMPUTER AIDED ENGINEERING DESIGN

̇rP() = () = P[ () – ()]

=0 =0 –1

u d –1 –1

du

Bu nB u B u

j

n

j j

n

j

n

j j

n

j

ΣΣn

= =0 () – () = () – ()

–1

–1

–1

=0

–1

–1

=0

–1

+1

–1

=0

–1

nBu Bun Bu Bu–1

j

n

j j

n

j

n

j j

n

j

n

j j

n

j

n

j j

⎡ΣΣPPΣ ΣP Pn

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎤

⎦

⎥

noting that Bu–1n–1() = 0.

rPP ̇() = ( – ) ()

=0

–1

+1

un B u–1

j

n

j j j

Σ n

Thus, ̇rPP(0) = (nn 10 – ) and (1) = (rPP ̇ n – n–1)

noting that only B 0 n–1(0) and Bnn–1–1(1) are 1.

(c) Geometry Invariance: Due to the partition of unity property of the Bernstein polynomials, the
shape of the curve is invariant under rotation and translation of the coordinate frame. This is illustrated
in Section 4.3 and shown as an example in Figure 4.16 (b).
(d) Convex Hull Property: The barycentric nature of Bernstein polynomials ensures that the Bézier
segment lies within the convex hull of the control points. The property is useful in intersection
problems, detection of interference, and provides estimates of the position of the curve by computing
the bounds of the polygon. Figure 4.17 shows an example.

Figure 4.17 Bézier curves and the associated convex hulls (closed convex polygons)

(a)

(b)

(c) (d)

(e) (f)

P 1

P 2

P (^0) P 3
P 1
P 2
P 0
P 3
P 1
P 2
P 0
P 3
P 2 P 1
P 0 P 3
P 1 P 2
P 0 P 3
P 1
P 2
P 0
P 4
P 4
P 3