Graphics Reference
In-Depth Information
-
1331
3630
3030
14 10
-
Ê
ˆ
Á
Á
Á
˜
˜
˜
-
=
()
M
s3
16
.
(11.82)
-
Ë
¯
Note how these formulas are compatible with the formulas in equation (11.68). The
derivative can be computed using
p
p
p
p
Ê
ˆ
i
-
1
Á
Á
Á
˜
˜
˜
¢
()
=
(
)
i
2
qu
3210
u u
M
.
(11.83)
i
s
3
i
+
1
Ë
¯
i
+
2
The points
p
j
in equations (11.78) and (11.81) are called the
ith B-spline coeffi-
cients
of the uniform curve p(u). One can also define the nonuniform B-splines with
matrices, but this takes more than one matrix. One needs separate matrices for com-
puting the curve near the endpoints. See [PokG89].
One important difference between clamped uniform and uniform B-spline curves
is that the former start at the first control point and end at the last one whereas the
latter do not. See Figure 11.18. As a partial demonstration of this point we analyze
the quadratic uniform case more closely. Using formula (11.78) we see that
()
=
()
(
)
q
i
012
1
-
pp
+
i
i
and
()
=
()
(
)
q
n
112
pp
,
+
-
1
n
-
1
n
that is, the curve starts at the midpoint of the segment [
p
0
,
p
1
], ends at the midpoint
of [
p
n-1
,
p
n
], and passes through the midpoints of all the other segments. Since
¢
()
=-
-
q
i
0
pp
i
i
1
p
1
p
1
p
2
p
2
p
5
p
5
p
4
p
4
p
3
p
3
p
0
p
0
(a)
(b)
Figure 11.18.
Clamped uniform versus uniform B-splines.
