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