Computer Aided Engineering Design

DESIGN OF SURFACES 243

The uniform knot vector is computed as [ti,i = 0,.. ., 7] = [0, 1, 2, 3, 4, 5, 6, 7]. A manual calculation
will be obviously lengthy. It can be conveniently done using any of the programming languages or
software with adequate graphics user interface. The surface is shown below in Figure 7.30(b).
(c) If some of the control points (e.g. the first row) are changed as given below, the change in
shape of the surface is shown in Figure 7.30(c).

rrrr

rrrr

rrrr

rrrr

rrrr

00 01 02 03

10 11 12 13

20 21 22 23

30 31 32 33

40 41 42 43

=

{0, 0, 3} {0, 1, 5} {0, 2, 5} {0, 2, 3}

{1, 0, 0} {1, 1, 2} {1, 2, 1} {1, 3, 2}

{2, 0, 0} {2, 1, 3} {2, 2, 2} {2, 3, 3}

{3, 0, 0} {3, 1, 2} {3, 2, 1} {3, 3, 2}

{4, 0, 0} {4, 1, 1} {4, 2, 1} {4, 2, 0}

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

(d) In another trial, a quadratic-cubic B-spline is created. Observe the change in shape Figure
7.30(d), for this quadratic-cubic B-spline surface for the same control points as in Figure 7.30(c).

7.5 Closed B-Spline Surface

Closed B-spline surface with uniform knot vectors can be created in a similar manner as described
in Exercises, Chapter 15 for creating closed B-spline curves.
Let the control points be rij (i = 0,.. ., n) and (j = 0,.. ., m). Let us restrict our discussion, for
simplicity, to a closed cubic uniform B-spline surface using 45 control points r 00 ,r 01 ,r 02 ,r 03 ,r 04 ,
r 05 ,r 06 ,r 07 ,r 00 ;r 10 ,r 11 ,r 12 ,r 13 ,r 14 ,r 15 ,r 16 ,r 17 ,r 10 ;r 20 ,r 21 ,r 22 ,r 23 ,r 24 ,r 25 ,r 26 ,r 27 ,r 20 ;r 30 ,r 31 ,
r 32 ,r 33 ,r 34 ,r 35 ,r 36 ,r 37 ,r 30 ;r 40 ,r 41 ,r 42 ,r 43 ,r 44 ,r 45 ,r 46 ,r 47 ,r 40. Observe that the last control point
in each segment is the same as the first. Uniform knot vector [0 1 2 3 4 5 6 7 8 9 10 11 12 13 14]
is to be used.
For a cubic B-spline surface, the equations of the surface segments can be written as

rUM

rr r r

rr r r

rr

1, + 1

2

0,( mod 8)+1 0,(( +1)mod 8)+1 0,(( + 2) mod 8)+1 0,(( +3) mod 8)+1

1,( mod 8)+1 1,(( +1)mod 8)+1 1,(( + 2) mod 8)+1 1,(( +3) mod 8)+1

2,( mod 8)+1 2,

(, ) =

1

j 6

jj j j

jj j j

j

u v ⎛

⎝

⎞

⎠ (((( +1)mod 8)+1 2,(( + 2) mod 8)+1 2,(( +3) mod 8)+1

3,( mod 8)+1 3,(( +1)mod 8)+1 3,(( + 2) mod 8)+1 3,(( +3) mod 8)+1

jj j

jj j j

T

rr

rr r r

MV

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

rUM

rr r r

rr r r

rr

2, +1

2

1,( mod 8)+1 1,(( +1)mod 8)+1 1,(( + 2) mod 8)+1 1,(( +3) mod 8)+1

2,( mod 8)+1 2,(( +1)mod 8)+1 2,(( + 2) mod 8)+1 2,(( +3) mod 8)+1

3,( mod 8)+1 3,

(, ) =

1

j 6

jj j j

jj j j

j

uv ⎛

⎝

⎞

⎠ (((( +1)mod 8)+1 3,(( + 2) mod 8)+1 3,(( +3) mod 8)+1

4,( mod 8)+1 4,(( +1)mod 8)+1 4,(( + 2) mod 8)+1 4,(( +3) mod 8)+1

jj j

jj j j

T

rr

rr r r

MV

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

(7.60)
Here,j∈ [0,.. ., 7]. The surface will be closed in v and open in u direction. This will be constituted
by (2 × 8) or 16 sliding surface segments. Matrices U,M and V are given by

UM = V

1

, =

–1 3 –3 1

3–630

–3 0 3 0

1410

, =

1

3

2

3

2

u

u

u

T

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

v

v

v