Graphics Reference
In-Depth Information
that of the following curve. This can be confirmed by differentiating the basis
functions (9.28)-(9.31):
3
t
2
+6
t
B
0
(
t
)=
−
−
3
(9.46)
6
B
1
(
t
)=
9
t
2
−
12
t
(9.47)
6
9
t
2
+6
t
+3
6
B
2
(
t
)=
−
(9.48)
B
3
(
t
)=
3
t
2
6
(9.49)
Evaluating (9.46)-(9.49) for
t
=0and
t
= 1, we discover the slopes 0
.
5
,
0
,
−
0
.
5
,
0 for the joins between B
3
,
B
2
,
B
1
,
B
0
.The
third level
of curve continu-
ity,
C
2
, ensures that the rate of change of slope at the end of one basis curve
matches that of the following curve. This can be confirmed by differentiating
(9.46)-(9.49):
B
0
(
t
)=
−
t
+ 1
(9.50)
B
1
(
t
)=3
t
−
2
(9.51)
B
2
(
t
)=
−
3
t
+ 1
(9.52)
B
3
(
t
)=
t
(9.53)
Evaluating (9.50)-(9.53) for
t
=0and
t
= 1, we discover the values 1
,
2
,
1
,
0
for the joins between B
3
,
B
2
,
B
1
,
B
0
. These combined continuity results are
tabulated in Table 9.4.
−
9.6.3 Non-Uniform B-Splines
Uniform B-splines are constructed from curve segments where the parameter
spacing is at equal intervals.
Non-uniform
B-splines, with the support of a
knot vector, provide extra shape control and the possibility of drawing periodic
shapes. Unfortunately an explanation of the underlying mathematics would
take us beyond the introductory nature of this text, and readers are advised
to seek out other topics dealing in such matters.
Table 9.4.
Continuity properties of cubic B-splines
t
t
t
C
0
0
1
1
0
1
C
2
0
1
B
3
(t)
B
3
(t)
B
3
(t)
0
1/6
0
0.5
0
1
B
2
(t)
B
2
(t)
B
2
(t)
1/6
2/3
0.5
0
1
−
2
B
1
(t)
B
1
(t)
B
1
(t)
2/3
1/6
0
−
0
.
5
−
2
1
B
0
(t)
B
0
(t)
B
0
(t)
1/6
0
−
0
.
5
0
1
0
