Graphics Reference
In-Depth Information
Closed Periodic B-Spline Curves
For closed periodic B-spline curves, we need to repeat some of the control
polygon vertices. The curve in this case is given by
m−
1
P
j
+1
(
s
)=
N
i
+1
,m
(
s
)
V
((
j
+
i
)
mod
(
n
+ 1)) + 1
0
≤
j
≤
n.
(5.23)
i
=0
In the matrix form, this can be written as
⎛
⎞
V
(
jmod
(
n
+1))+1
V
((
j
+1)
mod
(
n
+1))+1
.
V
((
j
+1+
n−m
)
mod
(
n
+1))+1
⎝
⎠
P
j
+1
(
s
)=(
S
)(
N
)
.
(5.24)
Here, mod is the modulo or remainder function, e.g., 5
mod
3 = 2 (mod is the
remainder function).
Example:
Find the 4th order closed B-spline curve whose control polygon is a square
with 8 vertices, say,
V
1
=(2
,
0)
,V
2
=(4
,
0)
,V
3
=(4
,
2)
,V
4
=(4
,
4)
,V
5
=
(2
,
4)
,V
6
=(0
,
4)
,V
7=(0
,
2), and
V
8
=(0
,
0).
Answer: Since the curve has order = 4, we have
m
= 4. The number of polygon
vertices is
n
= 8. Obviously here,
V
9
=
V
1
=(2
,
0), since the curve is closed.
The first curve segment,
P
1
from equation (5.24) is, therefore,
⎛
⎞
⎛
⎞
−
13
−
31
V
1
V
2
V
3
V
4
3
−
630
⎝
⎠
⎝
⎠
P
1
=(
S
)(
N
)
−
30 30
1410
⎛
⎞
20
40
42
44
⎝
⎠
=(
S
)(
N
)
.
From equation (5.21), the
S
matrix is
⎛
⎝
⎞
⎠
s
3
s
2
s
1
(
S
)=
,
and the reparameterized
N
matrix can be obtained from equation (5.22):
3
!
0
0
=
6
.
Therefore, the first segment can be computed using the following matrix equa-
tion;
1
N
1
,
4
=
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