Computer Aided Engineering Design

DESIGN OF CURVES 95

To maintain the same parametric (cubic) form, a linear relation between u and v is sought. An
advantage is that the directions of the tangent vectors at the ends are preserved. Thus,

v = αu + β, vi = αui + β and vj = αuj + β (4.18)

Here,vi and vj are values for v at the two ends of the trimmed segment. For re-parameterization,
vi = 0 and vj = 1 and hence from Eq. (4.18)

αβ =^1

( – )

and =

• uu –

u

ji uu

i

ji (4.19)

Letrv(v) represent the retained segment in Figure 4.6 noting that rv(v) in 0 ≤v≤ 1 is identical
tor(u) in ui≤u≤uj. Then

rv(0) = r(ui) and rv(1) = r(uj)

Figure 4.5 (b) Original (solid) and modified (dashed) C^2 continuous Ferguson curves

for Example 4.3. The shape change is global for a shift in data point

5

4

3

2

1

–1

–2

y

0123456

x

Figure 4.6 Trimming of a curve

D,u = 1

C,u = uj

v = 1

A,u = 0

B,u = ui

v = 0