Computer Aided Engineering Design

DESIGN OF CURVES 123

[x(t),y(t),z(t)] is termed as the rational Bézier segment. Note that Br tin() are barycentric, that is,

forwi≥0, the rational functions are all nonzero and they sum to 1. For a special case when wi = 1,
i = 0, ..., n, Eq. (4.66) yields a Bézier segment. An advantage when using rational Bézier segments
is the design freedom a user achieves by specifying weights wi to data points [xiyizi] at will. For
wi = 0, Pi(xi, yi,zi) has no effect on the shape of the curve since its corresponding coefficient Br tin()
is zero. As wi approaches infinity, all other Br tin() approach zero for which the curve converges to
Pi. A rational Bézier segment has all the properties of a Bézier segment. That is, a rational Bézier
segment passes through the end points, it lies within the convex hull defined by the control points
and it has the variation diminishing property. Further, by modifying weights appropriately, a rational
Bézier segment can be made more proximal to a chosen control point.

Example 4.10. For a set of control point P 0 = (4, 4), P 1 = (6, 8), P 2 = (8, 9) and P 3 = (10, 3) of
Example 4.6, compute the rational Bézier segment initially for all weights w 0 = w 1 = w 2 = w 3 = 1.
Alter the values of w 2 to realize the change in the curve shape.
Thex and y coordinates of points on the rational Bézier segment can be computed as

xt

wtxwttxwt txtx

wtwttwt tt

() =

(1 – ) + 3 (1 – ) + 3 (1 – ) +

(1 – ) + 3 (1 – ) + 3 (1 – ) +

0 3 01 2 122 2 3 3

0 3 1 2 223

yt

wtywttywt tyty

wtwttwt tt

() =

(1 – ) + 3 (1 – ) + 3 (1 – ) +

(1 – ) + 3 (1 – ) + 3 (1 – ) +

0

3

01

2

12

2

2

3

3

0

3

1

2

2

23

and the rational Bézier segments are shown in Figure 4.22 for w 0 = w 1 = w 3 = 1 and for different
valuesw 2. As w 2 is increased, the segment shapes towards P 2 = (8, 9).

Figure 4.22 Change in curve shape of a rational Bézier segment due to the change in weight

P 2

P 1

2

10

w 2 = 1

P 3

9 8 7 6 5 4 3

P 0

4 5 678 910

Another advantage when using rational Bézier segments is the ability to design conics precisely,
especially a circular arc which cannot be designed accurately using a polynomial Bézier segment of
even higher degrees like cubic, quadric or quintic. We consider designing conics using a rational
Bézier segment of degree 2 which is given as