Graphics Reference
In-Depth Information
4
3
2
1
2.5
3
3.5
4
4.5
5
FIGURE 3.4
Example of points produced by equal increments of an interpolating parameter for a typical cubic curve. Notice
the variable distance between adjacent points.
parametric values at equal-length steps along the curve. The first of these numeric methods constructs
the table by supersampling the curve and uses summed linear distances to approximate arc length. The
second numeric method uses Gaussian quadrature to numerically estimate the arc length. Both methods
can benefit from an adaptive subdivision approach to controlling error.
3.2.1
Computing arc length
To specify how fast the object is to move along the path defined by the curve, an animator may want to
specify the time at which positions along the curve should be attained. Referring to
Figure 3.5
as an
example in two-dimensional space, the animator specifies the following frame number and position
pairs: (0,
A
), (10,
B
), (35,
C
), and (60,
D
).
Alternatively, instead of specifying time constraints, the animator might want to specify the relative
velocities that an object should have along the curve. For example, the animator might specify that an
object, initially at rest at position
A
, should smoothly accelerate until frame 20, maintain a constant
B
A
time 10
time 0
C
D
time 35
time 60
FIGURE 3.5
Position-time pairs constraining the motion.
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