Game Development Reference
In-Depth Information
Figure 11.13 shows the same B-spline curve again, but this time with one of
its interior points replicated three times. The curve interpolates the replicated
control point, but only exhibits C continuity at that point. This is equivalent to
two separate B-spline curves for which the last control point of the first curve is
equal to the first control point of the second curve and each is replicated three
times.
11.6.2 B-Spline Globalization
()
Each piece
i t
of a uniform B-spline is defined over the range of parameter
Q
)
values
control points, we can define each piece
in terms of a global parameter u by assigning i
0,1
. For a curve having
n
+
1
t
[
=
i
and writing
t
()
(
)
Q
u
=
Q
u
t
.
(11.74)
i
i
i
()
The pieces
i u
compose the same curve using the range of parameter values
)
. We can write Equation (11.74) in terms of the B-spline basis func-
tions as follows.
∈−
1,
n
1
u
[
3
()
(
)
Q
u
=
B
u
t
P
(11.75)
i
k
i
i
+−
k
1
k
=
0
Any one of the control points P affects at most four pieces of the curve, and few-
er than four only if it occurs near the beginning or end of the sequence of control
points. For the piece
, the point P is weighted by the blending function B .
The same point is weighted by the blending function
i u
()
()
B for the piece
u
, the
i
+
1
()
u
blending function
B for the piece
, and the blending function
B for the
i
1
. Since the point P does not contribute to any other piece of the
curve, we can say that its weight is zero for any piece
()
u
piece
i
2
()
<−
i
2
u
where
or
j
j
>+
i
1
()
i N u that is al-
ways used as the weight for the point P for every piece of the curve. Since each
piece
. It is possible for us to construct a weighting function
j
()
)
()
i u
is defined over the parameter range
u
[
tt +
,
, we define
N
u as
i
i
1
i
(
)
)
B
ut
, f
u t t
[
,
;
0
i
+
1
i
+
1
i
+
2
(
)
[
)
Bu t
,
if
u tt
,
;
1
i
i
i
+
1
()
(
)
[
)
Nu
=
But
, f
u t t
, ;
(11.76)
i
2
i
1
i
1
i
(
)
[
)
B
ut
, f
u t t
,
;
3
i
2
i
2
i
1
0,
otherwise.
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