Computer Aided Engineering Design

244 COMPUTER AIDED ENGINEERING DESIGN

Example 7.15. A closed cubic B-spline surface is created using the following control points created
by selecting 5 sections (i = 0,.. ., 4) parallel to z-axis. Eight control points combined with the first
few points to close the circle create each circular cross section. The control points are given in
Tables 7.1 to 7.3.

Table 7.1

i ri 0 ri 1 ri 2 ri 3 ri 4 ri 5 ri 6 ri 7

0 (0 0 0) (–2 2 0) (–2 4 0) (0 6 0) (2 6 0) (4 4 0) (4 2 0) (2 0 0)

1 (0 0 1) (–2 2 1) (–2 4 1) (0 6 1) (2 6 1) (4 4 1) (4 2 1) (2 0 1)

2 (0 0 2) (–2 2 2) (–2 4 20 (0 6 2) (2 6 2) (4 4 2) (4 2 2) (2 0 2)

3 (0 0 3) (–2 2 3) (–2 4 3) (0 6 3) (2 6 3) (4 4 3) (4 2 30 (2 0 3)

4 (0 0 4) (–2 2 4) (–2 4 4) (0 6 4) (2 6 4) (4 4 4) (4 2 4) (2 0 4)

Table 7.2

i ri 0 ri 1 ri 2 ri 3 ri 4 ri 5 ri 6 ri 7

0 (0 0 0) (–2 2 0) (–2 4 0) (0 6 0) (2 6 0) (4 4 0) (4 2 0) (2 0 0)

1 (0 0 1) (–2 2 1) (–2 4 1) (0 6 1) (2 6 1) (4 4 1) (4 2 1) (2 0 1)

2 (0 0 2) (–1 1 2) (–1 2 2) (0 3 2) (2 3 2) (2 2 2) (2 1 2) (1 0 2)

3 (0 0 3) (–2 2 3) (–2 4 3) (0 6 3) (2 6 3) (4 4 3) (4 2 3) (2 0 3)

4 (0 0 4) (–2 2 4) (–2 4 4) (0 6 4) (2 6 4) (4 4 4) (4 2 4) (2 0 4)

Table 7.3

i ri 0 ri 1 ri 2 ri 3 ri 4 ri 5 ri 6 ri 7

0 (0 0 0) (–2 2 0) (–2 4 0) (0 6 0) (2 6 0) (4 4 0) (4 2 0) (2 0 0)

1 (0 0 1) (–2 2 1) (–2 4 1) (0 6 1) (2 6 1) (4 4 1) (4 2 1) (2 0 1)

2 (0 0 2) (–2 2 2) (–2 4 2) (0 6 2) (2 6 2) (8 8 2) (4 2 2) (2 0 2)

3 (0 0 3) (–2 2 3) (–2 4 3) (0 6 3) (2 6 3) (4 4 3) (4 2 3) (2 0 3)

4 (0 0 4) (–2 2 4) (–2 4 4) (0 6 4) (2 6 4) (4 4 4) (4 2 4) (2 0 4)

Data given in Table 7.1 generates the cylindrical B-spline surface shown in Figure 7.31(a). Data
given in Table 7.2 is generated by changing the control points (the row with i = 2) and corresponds
to Figure 7.31(b). Simlarly, by changing the control point r 25 (Table 7.3), Figure 7.31(c) is generated.
Observe the local changes in the surface due to change in control points.

7.6 Rational B-spline Patches (NURBS)

Analogous to Eq. (7.26), a B-spline surface patch can be defined as

rP(, ) = () ()

=0 =0 ,+ ,+

uNuN

i

m

j

n

vvΣΣ pp i qq j ij (7.61)

for an array {Pij,i = 0, ..., m;j = 0, ..., n} of control points. p and q are the orders of B-spline curves
along the u and v directions, respectively. From property 5.8.1B, the number of knots required in the
u direction is m + 1 + p while that along the v direction is n + 1 + q. Knot parameterization may be
performed along the two parametric directions using methods discussed in Section 5.10. More generally,
a rational B-spline patch may be computed as