Computer Aided Engineering Design

DESIGN OF CURVES 97

(12, 6) and the end tangents are given by Ti+2 = (–8, 5) and Ti+3 = (7, –7). Blend a curve between B
andC to ensure C^1 continuity.
From the given data, the geometric matrices G 1 andG 3 for curves (1) and (3) can be formed and
G 2 for the blending curve can be determined. Thus

GG 13 =

00

42

77

5–8

, =

84

12 6

–8 5

7–7

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

Therefore, G 2 =

42

84

(5) (–8)

(–8) (5)

αα

ββ

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

Coefficients α and β can be used to attenuate the magnitudes of the tangents while maintaining their
directions. Figure 4.8 shows two candidate blending curves for (α = 1, β = 1) and (α = 1, β = 4).

4.1.4 Lines and Conics with Ferguson Segments

The end points and tangents can be chosen such that one can generate curves of degree less than 3
with Hermite cubic curves. Recall from Eq. (4.7) that

r

P

P

T

T

( ) = [ 1] UMG

2–21 1

–3 3 –2 –1

0010

1000

(^32) =
+1
+1
uuuu
i
i
i
i
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
ForTi = Ti+1 = Pi+1 – Pi, we get

•

•

•

•

0 5 10 15

x(u)

7 6 5 4 3 2 1 0

–1

y(u)

α = 1, β = 4

α = β = 1

r(2)(u 2 )

r(3)(u 3 )

r(1)(u 1 )

Figure 4.8 Example 4.4 for curve blending