Interpolating Values - Computer Animation: Algorithms and Techniques - page 63

Complexity
The complexity of the underlying interpolation equation is of concern because this translates into com-
putational efficiency. The simpler the underlying equations of the interpolating function, the faster its
evaluation. In practice, polynomials are easy to compute, and piecewise cubic polynomials are the low-
est degree polynomials that provide sufficient smoothness while still allowing enough flexibility to
satisfy other constraints such as beginning and ending positions and tangents. A polynomial whose
degree is lower than cubic does not provide for a point of inflection between two endpoints; therefore,
it might not fit smoothly to certain data points. Using a polynomial whose degree is higher than cubic
typically does not provide any significant advantages and is more costly to evaluate.

Continuity
The smoothness in the resulting curve is a primary consideration. Mathematically, smoothness is deter-
mined by how many of the derivatives of the curve equation are continuous. Zero-order continuity
refers to the continuity of values of the curve itself. Does the curve make any discontinuous jumps
in its values? If a small change in the value of the parameter always results in a small change in
the value of the function, then the curve has zero-order, or positional, continuity. If the same can
be said of the first derivative of the function (the instantaneous change in values of the curve), then
the function has first-order ,or tangential , continuity . Second-order continuity refers to continuous cur-
vature or instantaneous change of the tangent vector (see Figure 3.2 ). In some geometric design envi-
ronments, second-order continuity of curves and surfaces may be needed, but often in animation
applications, first-order continuity suffices for spatial curves. As explained later in this chapter, when
dealing with time-distance curves, second-order continuity can be important.

Positional discontinuity at the
point

Positional continuity but not
tangential continuity at the point

circular arcs

Positional and tangential continuity
but not curvature continuity at the
point

Positional, tangential, and curvature
continuity at the point

FIGURE 3.2
Continuity (at the point indicated by the small circle).