Computer Aided Engineering Design

90 COMPUTER AIDED ENGINEERING DESIGN

respective independent parameters, that is, a point on the composite curve belongs to r(1)(u 1 ) for
0 ≤u 1 ≤ 1.
The composite curve is said to be positioncontinuous or C^0 continuous if the end point of the first
segmentr(1) and the start point of the second segment r(2) are coincident, that is

r(1)(u = 1) = r(2)(u 2 = 0) (4.10)
For slope orC^1 continuity at the junction point, the respective tangents of the two segments should
have the same direction (not necessarily the same magnitude) for which

αα 1 d (1)(1) = 2 (2)(0) =

du

d

du

rrt (4.11)

whereα 1 and α 2 are the normalizing scalars and t is the unit tangent vector at the junction point. In
Figure 4.4, r(1)(u 1 ) is a Ferguson segment with end points (xi,yi, zi), (xi+1,yi+1,zi+1) and end slopes

Figure 4.3 Change of shape in a Ferguson segment for different end tangent magnitudes

ci = 8.5

ci+1 = 15.2

40

30

20

8 7 6 5 4 3 2

y(u)

–10 12345678

x(u)

u 1 = 1

u 2 = 0

u 2 = 1

r(2)(u 2 )

r(1)(u 1 )

u 1 = 0

Pi(xi,yi,zi)

Ti(pi,qi,ri)

Pi+2(xi+2,yi+2,zi+2)

Ti+2(pi+2,qi+2,ri+2)

Pi+1(xi+1,yi+1,zi+1)

Ti+1(pi+1,qi+1,ri+1)

Figure 4.4 A composite Ferguson curve