92 COMPUTER AIDED ENGINEERING DESIGN
wheren + 1 is the number of data points. Eq. (4.17) suggests that for a cubic composite curve to
be curvature continuous throughout, the intermediate slopes Ti are related and thus need not be
specified. For given n + 1 data points Pi,i = 0, 1,... , n, Eq. (4.17) provides n – 1 relations, one
for each intermediate junction point, in n+ 1 unknown slopes Ti,i = 0, 1,... , n. Thus, two
additional conditions are required can be the slopes T 0 and Tn specified at the two ends of the
composite Ferguson’s curve. Once all the slopes are determined, a C^2 continuous Ferguson’s composite
curve is obtained.
Example 4.3. For data points A(0, 0), B(1, 2), C(3, 2) and D(6, –1), determine C^1 and C^2 continuous
Ferguson curves. For the first case, use slopes as 45°, 30°, 0° and – 45° at the data points. For a C^2
continuous curve, use end slopes as 45° and –45°, respectively. Comment on the variation in the
shapes of the composite curves if (a) data point C(3, 2) is relocated to (1.5, 4) and (b) slope at point
(0, 0) is modified to 90°.
Using chain rule,dy
dxdy du
dx du
=(/)
(/)
= tan ,θ where θ is the slope at a data point. Fordx
du
= 1, at alldata points, the following table summarizes the end tangent computation.
i Data point θ
dy
dudx
du
= tan θ Ti0 A(0, 0) 45 ° 1 (1, 1)1 B(1, 2) 30 °^1
31,
1
3⎛
⎝
⎜⎞
⎠
⎟2 C (3, 2) 0 ° 0 (1, 0)
3 D(6, – 1) –45° –1 (1, –1)Recursive use of Eq. (4.7) would result in the Ferguson segments between two successive data
points. For instance, the segment between A and B is
r 11 ( ) = [ 13 12 1] 112–21 1
–3 3 –2 –1
0010
100000
12
11
11/3uuuu , 0 1u⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥≤≤orr 11 (uu u) = [ 1 , –2.42 13 + 3.42uu 12 + 1 ]
Similarly, the curve segment BC isr 22 ( ) = [ 23 22 22 1]2–21 1
–3 3 –2 –1
0010
100012
32
11/3
10uuuu , 0 u 1⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥≤≤orr 22 (uuuu) = [– 2 23 + 3^22 + 2 + 1, 0.58(uuu^32 – 2 22 + ) + 2] 2
Finally, segment CD is