Graphics Reference
In-Depth Information
P
i
+3
P
i
+1
P
i
+7
S
i
+1
S
i
+2
S
i
+5
S
i
S
i
+3
S
i
+4
S
i
+6
P
i
+2
P
i
P
i
+6
P
i
+4
P
i
+8
Fig. 9.14.
The construction of a uniform non-rational B-spline curve.
A single segment
S
i
(
t
) of a B-spline curve is defined by
3
S
i
(
t
)=
P
i
+
r
B
r
(
t
)
for
0
≤
t
≤
1
(9.40)
r
=0
where
t
3
+3
t
2
t
)
3
B
0
(
t
)=
−
−
3
t
+1
=
(1
−
(9.41)
6
6
B
1
(
t
)=
3
t
3
6
t
2
+4
6
−
(9.42)
3
t
3
+3
t
2
+3
t
+1
6
B
2
(
t
)=
−
(9.43)
B
3
(
t
)=
t
3
6
(9.44)
These are the B-spline
basis functions
and are shown in Figure 9.15.
Although it is not apparent, these four curve segments are part of one
curve. The basis function B
3
starts at zero and rises to 0.1666 at
t
=1.Itis
taken over by B
2
at
t
= 0, which rises to 0.666 at
t
= 1. The next segment
is B
1
, which takes over at
t
= 0 and falls to 0.1666 at
t
= 1. Finally, B
0
takes over at 0.1666 and falls to zero at
t
= 1. Equations (9.28)-(9.31) are
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
Fig. 9.15.
The B-spline basis functions.
