164 COMPUTER AIDED ENGINEERING DESIGN
r
r
r
r
1 2
0
1
2
() =^12 [ 1]
1–21
–2 2 0
110
uuu
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
r
r
r
r
2 2
1
2
3
() =^1
2
[ 1]
1–21
–2 2 0
110
uuu
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
r
r
r
r
3 2
2
3
4
() =^12 [ 1]
1–21
–2 2 0
110
uuu
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
or r
r
r
r
i
i
i
i
() = uuu^1
2
[ 1]
1–21
–2 2 0
110
2
–1
+1
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
(b) Similarly, as above, show that the ith segment of a cubic B-spline curve is given by
r
r
r
r
r
i
i
i
i
i
() = uuuu^16 [ 1]
–1 3 –3 1
3–630
–3 0 3 0
1410
32
–1
+1
+2
⎛
⎝
⎜
⎜
⎜
⎜⎜
⎞
⎠
⎟
⎟
⎟
⎟⎟
⎛
⎝
⎜
⎜
⎜
⎜⎜
⎞
⎠
⎟
⎟
⎟
⎟⎟
- Using the above formulation, closed cubic B-spline curves can be generated. For example, let there be 7
control points r 0 ,r 1 ,r 2 , ......., r 6 (ri,i = 0, 1, 2, ..., 6). There will be n + 1 = 7 curve segments each of them
will be cubic B-spline, and can be written as
r
r
r
r
r
j
jn
jn
jn
jn
() = uuuu^1
6
[ 1]
–1 3 –3 1
3–630
–3 0 3 0
1410
32
( –1)mod( +1)
mod( +1)
( +1)mod( +1)
( +2)mod( +1)
⎛
⎝
⎜
⎜
⎜
⎜⎜
⎞
⎠
⎟
⎟
⎟
⎟⎟
⎛
⎝
⎜
⎜
⎜
⎜⎜
⎞
⎠
⎟
⎟
⎟
⎟⎟
Here, ‘mod’ is the “modulo” function which means that, if j = 2, j mod 7 = 2 (the remainder as a result of
this division).