164 COMPUTER AIDED ENGINEERING DESIGN
rr
r
r1 20
1
2() =^12 [ 1]1–21
–2 2 0
110uuu⎛⎝⎜
⎜
⎜⎞⎠⎟
⎟
⎟⎛⎝⎜
⎜
⎜⎞⎠⎟
⎟
⎟rr
r
r2 21
2
3() =^1
2
[ 1]1–21
–2 2 0
110uuu⎛⎝⎜
⎜
⎜⎞⎠⎟
⎟
⎟⎛⎝⎜
⎜
⎜⎞⎠⎟
⎟
⎟rr
r
r3 22
3
4() =^12 [ 1]1–21
–2 2 0
110uuu⎛⎝⎜
⎜
⎜⎞⎠⎟
⎟
⎟⎛⎝⎜
⎜
⎜⎞⎠⎟
⎟
⎟or rr
r
rii
i
i() = uuu^1
2
[ 1]1–21
–2 2 0
1102–1+1⎛⎝⎜
⎜
⎜⎞⎠⎟
⎟
⎟⎛⎝⎜
⎜
⎜⎞⎠⎟
⎟
⎟(b) Similarly, as above, show that the ith segment of a cubic B-spline curve is given byrr
r
r
rii
i
i
i() = uuuu^16 [ 1]–1 3 –3 1
3–630
–3 0 3 0
141032–1+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
rr
r
r
rjjn
jn
jn
jn() = uuuu^1
6
[ 1]–1 3 –3 1
3–630
–3 0 3 0
141032( –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).