Computer Aided Engineering Design

162 COMPUTER AIDED ENGINEERING DESIGN

weights in Eq. 5.46 to 1 yields the B-spline curve), discussion in this chapter pertaining to the design
of B-spline curves all apply to NURBS.

Example 5.10. For data points in Example 5.7, that is, (0, 0), (0, 1), (2, 3), (2.5, 6), (5, 2), (6, 0) and
(7, –3), design a rational B-spline curve with basis functions of order 4. First set all weights to 1.
Increase the weight w 3 corresponding to (2.5, 6) to visualize the shape change.
The example is solved using a uniform knot vector [0, 1, 2,.. ., 10) for an open rational spline and
the knot vector with multiple knots, that is, [0, 3, 3, 3, 4, 5, 6, 7, 7, 7, 10) for rational spline clamped
at both ends. NURBS curves are shown in Figure 5.23. Notice that in both cases, for w 3 = 0, P 3 = (2.5,
6) is not considered a part of the control polyline and the NURBS curves lie within the convex hull
of (0, 0), (0, 1), (2, 3), (5, 2), (6, 0) and (7, –3). With increase in w 3 , the curves get closer to
(2.5, 6).

2

1

10

w 3 = 0

2

1

10

w 3 = 0

6

4

2

0

–2

–4

02468

(a)

6

4

2

0

–2

–4

02468

(a)

Figure 5.23 (a) Open and (b) clamped rational B-spline curves

Exercises

1. Compute a quadratic B-spline basis function as a polynomial spline like in Example 5.2. Take the knot
   vector as [0, 1, 2, 3].


2. Verify the result obtained above using the divided differences table for truncated power series function
   ft t[ , ] = ( – )jjt t+^2.


3. A B-spline curve is defined as

bb() = tNti=0 (^) ,+()
n
Σ pp i i
(a)Explain and provide the full support interval for b(t)· (b) Demonstrate algebraically the local shape
control property if bj is relocated to bj + v. For what interval of t would the curve change in shape.

4. A first order basis function is defined as, say

Mti ttC

1, i i–1

() = – for t∈ [ti–1,ti]

or N1,i(t) = C[fort∈ [ti–1,ti]

What should be the value of C if (a) ti–1≠ti and (b) if ti–1 = ti.