Interpolating Values - Computer Animation: Algorithms and Techniques - page 106

v 2 v 3 v 1

1.0

v 1

Smoothing kernel superimposed over step
function

1/8 3/4 1/8

Areas of tent kernel under the different
step function values

1
8

3
4

1
8

V v 1

v 3

v 2

Computation of value smoothed by applying area
weights to step function values

FIGURE 3.45
Example of a tent-shaped smoothing filter.

2.2

2

4 6 8 10

1.8

1.6

1.4

FIGURE 3.46
Sample data smoothed with convolution using a tent kernel.

interpolated. At the endpoints, the step function can be arbitrarily extended so as to cover the kernel
function when centered over the endpoints. Often the first and last points must be fixed because of ani-
mation constraints, so care must be taken in processing these. Figure 3.45 shows how a tent kernel is used
to average the step function data; Figure 3.46 shows the sample data smoothed with the tent kernel.

Smoothing by B-spline approximation
Finally, if an approximation to the curve is sufficient, then points can be selected from the curve, and,
for example, B-spline control points can be generated based on the selected points. The curve can then
be regenerated using the B-spline control points, which ensures that the regenerated curve is smooth
even though it no longer passes through the original points.

3.4.4 Determining a path along a surface
If one object is to move across the surface of another object, then a path across the surface must be
determined. If start and destination points are known, it can be computationally expensive to find
the shortest path between the points. However, it is not often necessary to find the absolute shortest
path. Various alternatives exist for determining suboptimal yet reasonably direct paths.