Interpolating Values - Computer Animation: Algorithms and Techniques - page 84

Constant acceleration: parabolic ease-in/ease-out
To avoid the transcendental function evaluation while still providing for a constant speed interval
between the ease-in and ease-out intervals, an alternative approach for the ease function is to establish
basic assumptions about the acceleration that in turn establish the basic form that the velocity-time
curve can assume. The user can then set parameters to specify a particular velocity-time curve that
can be integrated to get the resulting distance-time function.
The simple case of no ease-in/ease-out would produce a constant zero acceleration curve and a
velocity curve that is a horizontal straight line at some value v 0 over the time interval from 0 to total
time, t total . The actual value of v 0 depends on the total distance covered, d total , and is computed using the
relationship

distance ¼ speedtime

so that

d total
t total
In the case where normalized values of 1 are used for total distance covered and total time, v 0 ¼

v 0 ¼

1

(see Figure 3.15 ) .
The distance-time curve is defined by the integral of the velocity-time curve and relates time and
distance along the space curve through a function S ( t ). Similarly, the velocity-time curve is defined by
the integral of the acceleration-time curve and relates time and velocity along the space curve.
To implement an ease-in/ease-out function, constant acceleration and deceleration at the beginning
and end of the motion and zero acceleration during the middle of the motion are assumed. The assump-
tions of beginning and ending with stopped positions mean that the velocity starts out at 0 and ends at 0.
In order for this to be reflected in the acceleration/deceleration curve, the area under the curve marked

a

acceleration 0.0

t

time

0.0 1.0

1.0

velocity

v

0.0

t

time

0.0 1.0

1.0

distance

s

0.0

t

time

0.0 1.0

FIGURE 3.15
Acceleration, velocity, and distance curves for constant speed.