Computer Aided Engineering Design

SPLINES 151

(Figure 5.14b) making the latter of multiplicity k=2, the total number of non-zero B-splines over
ti–4 gets reduced from three to two (p–k= 4 – 2) since N4,i– 3 gets eliminated from that set. Further,
forti–4 = ti–3 = ti–2 raising the multiplicity of ti–4 to k = 3, N4,i– 2 gets removed from the set of non-
zero splines (Figure 5.14(c)) leaving only N4,i– 1 (p–k= 4 – 3) in the set and for ti–4 = ti–3 = ti–2 =
ti–1 with k = 4 for ti–4 no (p–k= 4 – 4) non-zero splines over ti–4 exists (Figure 5.14(d)).
Note that due to G1,N4,i–1(t) would be position discontinuous at ti–1.

ti–7 ti–6 ti–5 ti–4 = ti–3 = ti–2 ti–1

(c)

N4,i–3N4,i–2 N 4 j–1

ti–7 ti–6 ti–5 ti–4 = ti–3 = ti–2 = ti–1

(d)

N4,i–3N4,i–2N4,i–1

N4,i–3 N4,i–2 N4,i–1 N4,i–3 N4,i–2 N4,i–1

ti–7 ti–6 ti–5 ti–4 ti–3 ti–2 ti–1

(a)

ti–7 ti–6 ti–5 ti–4 = ti–3 ti–2 ti–1

(b)

Figure 5.14 Schematic behavior of the normalized fourth order B-splines with increase in the knot
multiplicity; non-zero splines are shown using thick lines

5.8 B-Spline Curves: Definition

With an insight into the properties of normalized B-splines shape functions, we may now attempt to
design B-spline curves. Given n+1 control points b 0 ,b 1 ,... , bn and a knot vector T = {t 0 ,t 1 ,....
,tm}, the B-spline curve, b(t) of order p may be expressed as a weighted linear combination using the
normalized B-spline functions as

bb() = ()

=0 ,+

tNt

i

n

Σ pp i i (5.34)

This form of B-spline curve is very similar to a Bézier curve wherein the basis functions are the
Bernstein polynomials. The degree of the Bernstein basis functions is one less than the number of
control points for a Bézier segment. However, in case of B-spline curves, the degree of the basis
functions is an independent choice specified by the user. The number of knots (m+1) get determined
by the relation m = n + p with the total number of basis functions (n+1) being the same as the
number of control points (Eq. (5.34) and p being the order of the basis functions and hence the
curve. The spline in Eq. (5.34) is called an approximating spline as the curve usually does not pass
through the data points. However, a B-spline curve is more proximal to the control polyline than
a Bézier segment.
Though Eq. (5.34) is valid for all t in [−∞,∞],b(t) = 0 for t≤t 0 and t>tm. Thus, restricting the
parameter range in [t 0 ,tm) seems reasonable. A more restrictive range for t may be one in which full
support of the basis functions is achieved, that is, over any knot span [tj,tj+ 1 ) in [t 0 ,tm), atmost p B-