Computer Aided Engineering Design

SPLINES 157

barycentric property of B-spline basis functions that over the span [t 5 ,t 6 ), there are four non-
zero such functions, that is, N4,6,N4,7,N4,8 and N4,9 that sum to 1. At t = t 5 ,N4,9 = 0, implying that
N4,6(t 5 ) + N4,7(t 5 ) + N4,8(t 5 ) = 1. This is consistent with the knot multiplicity property (Section 5.7 G2)
that over a simple knot t 5 (k = 1), the number of non-zero basis functions are three (p–k = 4 –1).
Now, if knot t 4 is moved to t 5 raising the multiplicity of the latter to 2, the function N4,8(t 4 =t 5 )
becomes zero leaving N4,6(t 5 ) + N4,7(t 5 ) = 1. Further, if t 3 = t 4 = t 5 so that the multiplicity of t 5 is 3,
N4,7(t 3 = t 5 ) = 0. This implies that N4,6(t 5 ) = 1. Or, in other words, from Eq. (5.34), the B-spline curve
will pass through b 2 for t = t 5. In general, therefore, if ti+ 1 = ti+ 2 =,... , = ti+p–1 with ti+p–1 having
multiplicityp– 1, only one basis function Np,p+i will be non-zero over ti+p–1, and from the barycentric
property, Np,p+i will be 1, implying that the B-spline curve will pass through bi.

Example 5.8.Using control points, (4, – 4), (4, – 4), (4, – 4), (2, – 4), (0, 0), (2, 4), (4, 6), (8, 0),
(6, – 4), (4, – 4), (4, – 4), (4, – 4) for a cubic B-spline curve in Figure 5.19 (f), use knot multiplicity
to ensure that the curve passes through control points (0, 0) and (8, 0). Start with a uniform knot
sequenceti = i,i = 0,... , 15.
The range of full support for this example is [3, 12). To have the spline pass through (0, 0), which
is the fifth control point (i= 4), the knot t4+4–1 = t 7 = 7 should have multiplicity 3, that is, t 5 = t 6 =
t 7 = 7 (say). Otherwise, to have the spline pass through (8, 0) which is the eighth control point
(i= 7), t 10 should have multiplicity 3, that is, t 8 = t 9 = t 10 = 10 (say). The two results are shown in
Figure 5.21. Note the slope discontinuity at the two respective control points which is expected from

0 2468

x(t)

(b)

6

4

2

0

–2

–4

–6

y(t)

024 68

x(t)

(a)

6

4

2

0

–2

–4

–6

y(t)

Figure 5.21 A B-spline curve passing through desired intermediate control points

Figure 5.20 A schematic of cubic B-spline basis functions

N4,4 N4,5 N4,6 N4,7 N4,8 N4,9 N4,10

t 0 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10