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