Graphics Reference
In-Depth Information
First, we add the two fictitious control points to the table:
i
t
i
P
i
−
1 .8 ( , )
0
1
(1, 3)
1
1.2
(2, 3)
2
1.7
(3, 4)
3
2.5
(3, 6)
4
3.3
(3, 8)
Now, to evaluate the curve at a particular
t
-value like
t
=
2.6, for example,
we determine that
t
3
≤
t
4
. This means that we're in the
i
th segment, where
i
=
3. We'll therefore need to evaluate
p
(
3, 1
)(
t
)
,
2.6
≤
...
,
p
(
3, 4
)(
t
)
.
p
(
3, 1
)(
t
)=
−
(
t
−
t
3
)(
t
−
t
4
)
2
t
4
)
2
, so
(22.15)
(
t
4
−
t
2
)(
t
3
−
0.8
2
p
(
3, 1
)(
2.6
)=
−
0.1
·
(
0.8
)
2
.
(22.16)
1.6
·
In evaluating the other three polynomials, the expressions
t
t
appear
repeatedly. To write efficient code, you'd want to evaluate these just once and
reuse them often.
−
t
i
and
t
i
+
1
−
Suppose that you're doing an animation in which you have a moving object,
and you want it at position
P
i
at time
t
i
for
i
=
. . .; the Catmull-Rom spline is
a natural choice. Now suppose you've got an object that's controlled by some
parameter such as a pinwheel whose rotation is specified in degrees at several
key times—say,
R
(
0
)=
45,
R
(
1
)=
360, and
R
(
3
)=
720—and you want to pro-
vide values for
R
at intermediate times. Again, the Catmull-Rom spline is a natu-
ral choice. Note, however, that if the rotations were specified by
R
(
0
)=
R
(
1
)=
0
and
R
(
3
)=
90, then the Catmull-Rom interpolant, at
t
=
0.5, would be
negative:
The pinwheel would start to spin backward before rushing to spin forward. While
this might give the feeling of anticipation of motion in a cartoon-like animation,
it would be inappropriate for an animation that was supposed to be physically
realistic.
Cubic B-splines (there are also linear, quadratic, quartic, etc., B-splines, but cubics
are widely used) are similar to Catmull-Rom splines. The key differences are that
the cubic B-spline (a) is
C
2
smooth, that is, both its first and second derivatives are
continuous functions, and (b) is noninterpolating, that is, it passes
near
the control
points, but not
through
them in general.
Cubic B-splines come in two flavors: uniform and nonuniform. We'll start
with the uniform B-spline. The formula for the cubic B-spline with control points
P
0
,
...
,
P
n
is
