DESIGN OF SURFACES 241Example 7.13 (Closed Bézier Surface). Closing the polyhedron formed by the control points can
create closed Bézier surfaces. It is required to create a closed tubular Bézier surface by using 5
control points (last control point being the same as the first) at each of the sections, the control point
net created by using 5 such sections. The data set is given below.
(a) P1 = [1, 0, 0; –2, 0, 3; 1, 0, 6; 2, 0, 3; 1, 0, 0], P2 = [1, 2, 0; –3, 2, 4; 1, 2, 8; 3, 2, 4; 1, 2, 0],
P3 = [1, 5, 0; –5, 5, 5; 1, 5, 10; 5, 5, 5; 1, 5, 0], P4 = [1, 7, 0; –4, 7, 3; 1, 7, 6; 4, 7, 3; 1, 7, 0],
P5 = [1, 9, 0; –2, 9, 2; 1, 9, 6; 1, 9, 2; 1, 9, 0]
In the following parts (b) and (c), the control points are changed. Show the effect of the changed
control points on the shape of the surface.
(b) P1 = [1, 0, 0; –2, 0, 3; 1, 0, 6; 2, 0, 3; 1, 0, 0], P2 = [1, 2, 0; –4, 2, 3; 1, 2, 6; 4, 2, 3; 1, 2, 0],
P3 = [1, 5, 0; –7, 5, 3; 1, 5, 6; 7, 5, 3; 1, 5, 0], P4 = [1, 7, 0; –9, 7, 3; 1, 7, 6; 9, 7, 3; 1, 7, 0],
P5 = [1, 9, 0; –11, 9, 3; 1, 9, 6; 11, 9, 3; 1, 9, 0]
(c) P1 = [1, 0, 0; –2, 0, 3; 1, 0, 6; 2, 0, 3; 1, 0, 0], P2 = [1, 2, 0; –2, 2, 3; 1, 2, 6; 2, 2, 3; 1, 2, 0],
P3 = [1, 5, 0; –2, 5, 3; 1, 5, 6; 2, 5, 3; 1, 5, 0], P4 = [1, 7, 0; –2, 7, 3; 1, 7, 6; 2, 7, 3; 1, 7, 0],
P5 = [1, 9, 0; –2, 9, 3; 1, 9, 6; 2, 9, 3; 1, 9, 0]
7.4 B-Spline Surface Patch
A B-spline surface patch can be created using the above definitions of the B-spline curves made in
Chapter 5 and the tensor product definition of the surface.
If uniform (periodic) knot vector is used the surfaces can be easily constructed using the following
equations (refer to the problems in Exercises, Chapter 5).
Uniform quadratic B-spline surface
rrrr
rrr
rrr(, ) =1
2
[ 1]1–21
–2 2 0
1101–21
–2 2 0
110 12
200 01 02
10 11 12
20 21 222uuuTvv
⎛ v
⎝⎞
⎠⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥(7.58)Uniform cubic B-spline surface
ru(, ) = u u u1
6
[ 1]–1 3 –3 1
3–630
–3 0 3 0
1410–1 3 –3 1
3–630
–3 0 32
3200 01 02 03
10 11 12 13
20 21 22 23
30 31 32 33v ⎛
⎝⎞
⎠⎡⎣⎢
⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥
⎥rrrr
rrrr
rrrr
rrrr00(^14101)
3
2
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
T
v
v
v
(7.59)
Example 7.14. (Uniform Quadratic B-spline Surface). (a) A surface patch is to be constructed using
the matrix formulation for uniform quadratic B-spline surface patch given by
r
rrr
rrr
rrr
(, ) =
1
2
[ 1]
1–21
–2 2 0
110
1–21
–2 2 0
110 1
2
2
00 01 02
10 11 12
20 21 22
2
uuu
T
v
v
⎛ v
⎝
⎞
⎠
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥