Computer Aided Engineering Design

DESIGN OF CURVES 121

four data points can be chosen freely. For the second segment, three of the four points, namely q 0 ,q 1
andq 2 are constrained by the three continuity conditions; q 0 becomes the fourth point p 3 of the first
segment,q 1 is constrained to be placed along the vector p 2 p 3 , and q 2 must be placed on a plane
defined by the four data points previous to it. It is only the fourth point q 3 of the subsequent segment
that can be chosen freely. Note that different values of scalars λ and μ may be specified to choose q 1
andq 2 to satisfy slope and curvature continuities. Nevertheless, this freedom is indirect. This restricts
the flexibility in curve design for which reason, designers tend to prefer degree 5 or 7 Bézier
segments. When working with degree 3 segments, if a user seeks more flexibility in design, subdivision
(Section 4.4.2) or degree elevation (section 4.4.3) can be incorporated to generate more data points.

Example 4.9. For a two segment C^2 continuous composite Bézier curve, data points for the first cubic
segment are given as p 0 (0, 0, 0), p 1 (1, 2, 0), p 2 (3, 2, 0) and p 3 (6, –1, 0). Generate the second cubic
segment with some chosen values of scalars λ and μ as they appear in Eqs. (4.59) and (4.62).
Let the data points for the second cubic segment be q 0 ,q 1 ,q 2 and q 3. For position continuity
(Eq. 4.58), q 0 ≡p 3 = (6, –1, 0). For slope continuity from Eq. (4.59), we have

q 1 = (λ + 1)p 3 – λp 2 = (λ + 1) (6, – 1, 0) – λ(3, 2, 0) = (6 + 3λ, –1 –3λ, 0)

while for curvature continuity, using Eq. (4.64) yields

qpppp 2

2

= 1 – 2 + 2 ( 32 – ) – ( )( 21 – )

μλ

λλ

⎛

⎝

⎜

⎞

⎠

⎟

+ 1 –

2

+ 2 [3, – 3, 0] – ( ) [2, 0, 0] + [6, – 1, 0]

μλ^2

λλ

⎛

⎝

⎜

⎞

⎠

⎟

= 9 + 4 –

3

2

, – 4 – 6 +^3

2

⎛ λμλ (^22) λμλ , 0
⎝
⎞
⎠
q 3 being a free choice, it is assumed as (5, 2, 2) in this example. Figure 4.21 shows the composite
Bézier curve for different values of the pair (λ,μ). Note that five control points around the junction
point and including it lie on the x-y plane.

4.6 Rational Bézier Curves

Bézier segments, by themselves, do not have any local control in that change in the position of a data
point causes the shape of the entire segment to change. Achieving local shape control is the prime
motivation to discuss B-spline curves in Chapter 5. However, in this section, we discuss Rational
Bézier curves that provide more freedom to a designer in defining the shape of a segment/curve.
In Chapter 2, homogenous coordinates were introduced that helped in unifying rotation and
translation as matrix multiplication opertions. In essence, Pi≡ [xi,yi,zi, 1] and PiH≡ [Xi = wiyi,Yi =
wizi,Zi = wizi,Wi = wi] represent the same pooint in the Euclidean space E^3 .Pi is, in a way, the
projection of PiH on the wi = 1 hyperplane. Since a curve (surface or solid) may need to be gtransformed
at some intermediatge stage in a design operation, it behooves to work with the generalized homogenous
coordinates of data points. This provides more freedom to a designer in that the user now needs to
specify weight wi as an additional parameter with the Euclidean coordinates [xi, yi,zi] of a data point.
With n + 1 data points PiH,i = 0, ..., n, the nth degree Bézier segment can be defined as