Computer Aided Engineering Design

154 COMPUTER AIDED ENGINEERING DESIGN

(D) b(t) is Cp-k–^1 continuous at a knot of multiplicity k
Ift = ti, a knot of multiplicity k, since Np,i(t) is Cp–k–^1 continuous, so is the curve b(t) at that knot.

02468

x(t)

6

4

2

0

–2

–4

y(t)

Figure 5.18 Local shape modification of B-spline curves

0246 8

x(t)

6

4

2

0

–2

–4

y(t)

Figure 5.17 The convex hull property of

B-spline curves

For any other twhich is not a knot, the B-spline
curve is a polynomial of order p and is infinitely
differentiable.

(E) Variation diminishing property
The variation diminishing property discussed
previously for Bézier segments also holds for B-
spline curves. This feature along with the strong
convex hull property helps predict the shape of
B-spline curves better than that of Bézier
segments.

(F) Local modification scheme: Relocating bi
only affects the curve b(t) in the interval
[ti,ti+p)
This follows from the local support property in
Section 5.7(C) of B-spline basis functions. Let
the control point bibe moved to a new position
bi + v. Then, the new B-spline curve,c(t) from Eq. (5.34) is

cbbvb() = () + ()( + ) + ()

=0

–1

tNtNtk ,+ ,+ =+1Nt,+

i

pp k k pp i i ki

n

ΣΣpp k k

= () + () = () + ()

=0 ,+ ,+ ,+

Σ

k

n

Nt Ntpp k bvb vk pp i tNtpp i (5.35)

The coefficient of v, i.e., Np,i+p(t) is non-zero in [ti,ti+p). For t is not in this interval, Np,i+p(t)v has
no effect on the shape of b(t). However, for t∈ [ti,ti+p),Np,i+p(t) is non-zero and the curve b(t) gets
locally modified by Np,i+p(t)v. To show this, the data point (5, 2) in Example 5.7 is moved to a new
location (8, 6) for which the local change in the open spline in Figure 5.16 (a) is shown in Figure 5.18
(dotted lines).