Interpolating Values - Computer Animation: Algorithms and Techniques - page 62

several sections address these issues of generating points along a path defined by control points and
distributing the points along the path according to timing considerations.

3.1.1 The appropriate function
Appendix B.5 contains a discussion of various specific interpolation techniques. In this section, the
discussion covers general issues that determine how to choose the most appropriate interpolation tech-
nique and, once it is chosen, how to apply it in the production of an animated sequence.
The following issues need to be considered in order to choose the most appropriate interpolation
technique: interpolation versus approximation, complexity, continuity, and global versus local control.

Interpolation versus approximation
Given a set of points to describe a curve, one of the first decisions an animator must make is whether the
given values represent actual positions that the curve should pass through ( interpolation ) or whether
they are meant merely to control the shape of the curve and do not represent actual positions that the
curve will intersect ( approximation ) (see Figure 3.1 ) . This distinction is usually dependent on whether
the data points are sample points of a desired curve or whether they are being used to design a new
curve. In the former case, the desired curve is assumed to be constrained to travel through the sample
points, which is, of course, the definition of an interpolating spline . 1 In the latter case, an approximat-
ing spline can be used as the animator quickly gets a feel for how repositioning the control points influ-
ences the shape of the curve.
Commonly used interpolating functions are the Hermite formulation and the Catmull-Rom spline.
The Hermite formulation requires tangent information at the endpoints, whereas Catmull-Rom uses
only positions the curve should pass through. Parabolic blending, similar to Catmull-Rom, is another
useful interpolating function that requires only positional information. Functions that approximate
some or all of the control information include Bezier and B-spline curves. See Appendix B.5 for a more
detailed discussion of these functions.

An approximating spline in which only the
endpoints are interpolated; the interior control
points are used only to design the curve

An interpolating spline in which the spline
passes through the interior control points

FIGURE 3.1
Comparing interpolation and approximating splines.

1 The term spline comes from flexible strips used by shipbuilders and draftsmen to drawsmooth curves. In computer graphics, it
generally refers to a wide class of interpolating or smoothing functions.