Computer Aided Engineering Design

DESIGN OF CURVES 109

The transformed Bézier segment is plotted in Figure 4.16 (b). Observe that shape of the segment
does not change. This is due to the affine properties of Bernstein polynomials as weighting functions.
Modified curves for P 2 relocated to (12, 12) and P 1 repositioned at (3, 10) are shown in Figure 4.16
(c) comparing with the original curve. Also note that moving a single data point affects an overall
change in the curve segment.

Figure 4.16 Bézier segments for Example 4.6

(c) Global change affected when control points are relocated

12

10

8

6

4

2

y

2 4 6 8 10 12

x

10

5

0

–5

y

0510

x

(b) Transformation does not change

the shape of the segment

4567 8910

x

(a) Bézier segment

9 8 7 6 5 4 3

y

P 1

P 2

P 0

P 3

P 0

P 1

P 2

P 3

P 1 ′

P 0 ′

P 3 ′

P 2 ′

P 0

P 1 P 2

P 3

P 1 ′

P 2 ′

4.4.1 Properties of Bézier Segments

Based on the properties of Bernstein polynomials, much can be known about the Bézier segments.
These properties are discussed for a general Bézier segment of degree n.
(a) End Points: Note that at u = 0, B 0 n(0) = 1 while all the other polynomials Bin(0) are zero from
the non-negativity and partition of unity properties of Bernstein polynomials. Thus, P 0 is an end point
on the Bézier segment. Also, at u = 1, Bnn(1) = 1 while all other Bernstein coefficients are zero,
implying that Pn is another end point on the segment.
(b) End Tangents: The end tangents have the directions of P 1 – P 0 and Pn – Pn–1, respectively.
From Eq. (4.38),