Interpolating Values - Computer Animation: Algorithms and Techniques - page 98

While the Frenet frame provides useful information about the curve, there are a few problems with
using it directly to control the orientation of a camera or object as it moves along a curve. First, there is
no concept of “up” inherent in the formation of the Frenet frame. The v vector merely lines up with the
direction of the second derivative. Another problem occurs in segments of the curve that have no cur-
vature ( P 00 ( u )

0) because the Frenet frame is undefined. These undefined segments can be dealt with
by interpolating a Frenet frame along the segment from the Frenet frames at the boundary of the seg-
ment. By definition, there is no curvature along this segment, so the boundary Frenet frames must differ
by only a rotation around w. Assuming that the vectors have already been normalized, the angular dif-
ference between the two can be determined by taking the arccosine of the dot product between the two v
vectors so that y ¼

¼

acos( v 1

v 2 ). This rotation can be linearly interpolated along the no-curvature

segment ( Figure 3.33 ).
The problem is more difficult to deal with when there is a discontinuity in the curvature vector as is
common with piecewise cubic curves. Consider, for example, a curve composed of two semicircular
segments positioned so that they form an S ( sigmoidal ) shape. The curvature vector, which for any
point along this curve will point to the center of the semicircle that the point is on, will instantaneously
switch from pointing to one center point to pointing to the other center point at the junction of the two
semicircular segments. In this case, the Frenet frame is defined everywhere but has a discontinuous
jump in orientation at the junction (see Figure 3.34 ) .
However, the main problem with using the Frenet frame as the local coordinate frame to define the
orientation of the camera or object following the path is that the resulting motions are usually too
extreme and not natural looking. Using the w -axis (tangent vector) as the view direction of a camera
can be undesirable. Often, the tangent vector does not appear to correspond to the direction of “where

u j

v i u i

v j

w i

w j

Frenet frames on the boundary of an undefined Frenet frame
segment because of zero curvature

v i

u i u j

v j

The two frames sighted down the (common) w vector

u j

v i

u i

θ

θ

v j

The two frames superimposed to identify angular difference

FIGURE 3.33
Interpolating Frenet frames to determine the undefined segment.