Computer Aided Engineering Design

150 COMPUTER AIDED ENGINEERING DESIGN

of knots. The first normalized spline on the knot set [t 0 ,tm) is Np,p(t) while the last spline on this
set is Np,m(t) making a total of m– p + 1 basis splines. Letting n + 1 = m– p + 1 gives the result
(m = n + p).

(G) Multiple knots: Some knots in a given knot span may be equal for which some knot spans
may not exist

If a knot ti appears k times (i.e.ti–k+1 = ti–k+2 =... = ti), where k > 1, tiis called a multiple knot or
a knot of multiplicity k. Otherwise, for k = 1, ti is termed as a simple knot. Multiple knots significantly
change the properties of basis functions and are very useful in the design of B-spline curves. We may
note here that to ensure right continuity of Mk,i(t) and Nk,i(t) in case k consecutive knots coincide,

one assumes^0
0 = 0
convention when computing B-spline basis functions. If ti–1 = ti, then M1,i(t) and

N1,i(t) are defined as zero. Some properties of normalized B-splines with multiple knots are as
follows.

G1: At a knot i of multiplicity k, the basis function Np, i(t) is Cp–1–k continuous at that knot

Example 5.6.Verify using plots the discontinuity property above for quadratic normalized B-splines
using the parent knot sequence [0 1 2 3].
The plots for N3,i(t) using Eq. (5.33) are shown in Figure 5.13 for the knot sequence:
(a) [ti–3,ti–2,ti–1,ti]≡ [0 1 2 3], (b) [0 1 3 3] (k = 2 at ti= 3) and (c) [0 3 3 3] (k = 3 at ti= 3). For knots
with multiplicity 1, N3,i(t) is expected to be C^1 continuous everywhere, especially at the knot value
3 where the slope is zero. Raising knot multiplicity by one at knot value 3 results in slope discontinuity

as for t→ (^3) – , the slope is non-zero while for t→ (^3) +, the slope is zero. Thus, the B-spline is C^0
continuous at t = 3. Further increase in knot multiplicity by 1 at knot value 3 makes N3,i(t) position
discontinuous since for t→ (^3) −,N3,i(t) = 1 while for t→ (^3) +,N3,i(t) = 0.
k = 3
k = 1
k = 2
01234
t
1
0.8
0.6
0.4
0.2
0
N3,i(t)
Figure 5.13 Performance of N3,i(t) as knot multiplicity k of the knot at 3 is increased
G2. Over each internal knot of multiplicity k, the number of non-zero order pbasis functions
is at most p−−−−− k
The property is elucidated using normalized order 4 (cubic) B-splines. Figure 5.14(a) shows three
such splines, that is, N4,i– 3 ,N4,i– 2 and N4,i– 1 over the knot span [ti–7,ti–1) which are non-zero over a
simple knot ti–4 concurring with the property for p = 4 and k = 1. If knot ti– 3 is moved to ti– 4