Interpolating Values - Computer Animation: Algorithms and Techniques - page 80

There are various ways in which the distance-time function can be specified. It can be explicitly
defined by giving the user graphical curve-editing tools. It can be specified analytically. It can also
be specified by letting the user define the velocity-time curve, or by defining the acceleration-time
curve. The user can even work in a combination of these spaces. In the following discussion, the com-
mon assumption is that the entire arc length of the curve is to be traversed during the given total time.
Two additional optional assumptions (constraints) that are typically used are listed below. In certain
situations, it might be desirable to violate one or both of these constraints.

1. The distance-time function should be monotonic in t , that is, the curve should be traversed without
backing up along the curve.
2. The distance-time function should be continuous, that is, there should be no instantaneous jumps
from one point on the curve to a nonadjacent point on the curve.

Following the assumptions stated above, the entire space curve is to be traversed in the given total
time. This means that 0.0 ¼ S (0.0) and total_distance ¼ S ( total_time ). As mentioned previously, the

distance-time function may also be normalized so that all such functions end at 1.0

¼ S (1.0). Normal-
izing the distance-time function facilitates its reuse with other space curves. An example of an analytic
definition is S ( t )

t )* t (although this does not have the shape characteristic of ease-in/ease-out
motion control; see Figure 3.10 ) .

¼

(2

3.2.3 Ease-in/ease-out
Ease-in/ease-out is one of the most useful and most common ways to control motion along a curve.
There are several ways to incorporate ease-in/ease-out control. The standard assumption is that the
motion starts and ends in a complete stop and that there are no instantaneous jumps in velocity
(first-order continuity). There may or may not be an intermediate interval of constant speed, depending
on the technique used to generate the speed control. The speed control function will be referred to as s ¼
ease ( t ), where t is a uniformly varying input parameter meant to represent time and s is the output
parameter that is the distance (arc length) traveled as a function of time.

1

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1

Time

FIGURE 3.10
Graph of sample analytic distance-time function (2 t)*t.