Interpolating Values - Computer Animation: Algorithms and Techniques - page 86

must be covered during the time interval; the velocity is equal to the area below (above) the acc ( dec )
segment in Figure 3.16 . In the case of normalized time and normalized distance covered, the total time
and total distance are equal to 1. The total distance covered will be equal to the area under the velocity
curve ( Figure 3.17 ) . The area under the velocity curve can be computed as in Equation 3.21 .

1 ¼ A acc þ A constant þ A dec

1
2 v 0 t 1 þ v 0 ðt 2 t 1 Þþ

1
2 v 0 ð

(3.21)

1 ¼ 1 t 2 Þ

Because integration introduces arbitrary constants, the acceleration-time curve does not bear any
direct relation to total distance covered. Therefore, it is often more intuitive for the user to specify ease-
in/ease-out parameters using the velocity-time curve. In this case, the user can specify two of the three
variables, t 1 , t 2 , and v 0 , and the system can solve for the third in order to enforce the “total distance
covered” constraint. For example, if the user specifies the time over which acceleration and deceler-
ation take place, then the maximum velocity can be found by using Equation 3.22 .

2
ðt 2 t 2 þ

v 0 ¼ (3.22)

1 Þ

The velocity-time function can be integrated to obtain the final distance-time function. Once again,
the integration introduces an arbitrary constant, but, with the assumption that the motion begins at the
start of the curve, the constant is constrained to be 0. The integration produces a parabolic ease-in
segment, followed by a linear segment, followed by a parabolic ease-out segment ( Figure 3.18 ) .
The methods for ease control based on one or more sine curve segments are easy to implement and use
but are less flexible than the acceleration-time and velocity-time functions. These last two functions allow
the user to have even more control over the final motion because of the ability to set various parameters.

3.2.4 General distance-time functions
When working with velocity-time curves or acceleration-time curves, one finds that the underlying
assumption that the total distance is to be traversed during the total time presents some interesting
issues. Once the total distance and total time are given, the average velocity is fixed. This average

d

1

t 2

d v 0 2 t 1 0.0 t t 1

0.8

t 1

0.6

d v 0 t 1 t t 2

v 0 ( t t 1 )

2

0.4

t t 2

v 0

1 t 2

t 1

0.2

d v 0 v 0 ( t 2 t 1 ) v 0 ( t t 2 ) t 2 t 1.0

2 2

s

0.2 0.4 0.6 0.8 1

Time

FIGURE 3.18
Distance-time function for constant acceleration.