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a
(t) = 2
c
2
+ 6
c
3
t
= 2(3
b
0
− 6
b
1
+ 3
b
2
) + 6(−
b
0
+ 3
b
1
− 3
b
2
+
b
3
)t
Acceleration of a cubic
Bezier curve
= (6
b
0
− 12
b
1
+ 6
b
2
) + (−6
b
0
+ 18
b
1
− 18
b
2
+ 6
b
3
)t.
At the endpoints, the acceleration is given by
a
(0) = (6
b
0
− 12
b
1
+ 6
b
2
) + (−6
b
0
+ 18
b
1
− 18
b
2
+ 6
b
3
)0
Acceleration of a cubic
Bezier curve at the
endpoints
− 12
b
1
+ 6
b
2
,
a
(1) = (6
b
0
= 6
b
0
− 12
b
1
+ 6
b
2
) + (−6
b
0
+ 18
b
1
− 18
b
2
+ 6
b
3
)1
= 6
b
1
− 12
b
2
+ 6
b
3
.
As expected, the acceleration at the start is completely determined by the
first three control points, and the acceleration at the end is determined by
the last three control points.
Let's define
d
i
=
b
i+1
−
b
i
as shorthand for the delta between consecu-
tive control points, the vector of the ith edge of the Bezier control polygon.
With this notation, the acceleration formulas bear a striking resemblance
to the velocity formulas:
a
(0) = 6
b
0
− 12
b
1
+ 6
b
2
= 6
b
0
− 6
b
1
− 6
b
1
+ 6
b
2
= 6 ((
b
2
−
b
1
) − (
b
1
−
b
0
))
= 6(
d
1
−
d
0
),
Acceleration of a cubic
Bezier curve at the
endpoints, in terms of
the delta between
consecutive control
points
(13.36)
a
(1) = 6
b
1
− 12
b
2
+ 6
b
3
= 6
b
1
− 6
b
2
− 6
b
2
+ 6
b
3
= 6 ((
b
3
−
b
2
) − (
b
2
−
b
1
))
= 6(
d
2
−
d
1
).
(13.37)
The above discussion applies to Bezier curves of any degree. In general,
the pattern is this: if we move control point
b
i
, we affect the ith derivative
and higher at the start of the curve, but not lower-numbered derivatives.
(Similar statements apply at the end of the curve, regarding control point
b
i
and the derivative n−i and higher.) Of course, for a cubic spline that's just
about the end of the story, since we cannot move any control point without
potentially changing the third derivative at every point on the spline, since
the third derivative is constant for a cubic, and all higher derivatives are
zero. We come back to these ideas in
Section 13.8.1
when we talk about
the continuity conditions of two or more Bezier curve segments joined in a
spline.
13.5
Subdivision
Beginning with
Section 13.6,
this chapter addresses the topic of joining
together curves into a spline, which we can make as long and as complex as
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