Computer Aided Engineering Design

242 COMPUTER AIDED ENGINEERING DESIGN

Here,

rrr

rrr

rrr

00 01 02

10 11 12

20 21 22

=

{0, 0, 0} {0, 1, 0} (0, 2, 0}

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

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

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

With some care, the equation of surface can be manually derived. The surface is shown in Figure 7.30(a).
(b) A (quadratic-quadratic) B-spline surface patch is to be constructed using the following control
points, without use of the matrix formulation (using the original definition of the B-spline blending
functions and uniform knot vector)

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

{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}

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

0.5

0.4

0.3

0.5

0.75

1

1.25

1.5 0.5

0.75

1

1.25

1.5

2.25

2

1.75

1.5

3

1

2.5

1.5

0.5

1

1.5

2

2

2.5

3

2

1

0

3 2 1 0 0 1 2 3 4

3

2

1

0

3 2 1 0 0 1 2 3 4

(a)

(b)

(c) (d)

Figure 7.30 B-spline surfaces