Interpolating Values - Computer Animation: Algorithms and Techniques - page 97

Use of the Frenet frame
If an object or camera is following a path, then its orientation can be made directly dependent on the
properties of the curve. The Frenet frame 4 can be defined along the curve as a moving coordinate sys-
tem, ( u , v , w ), determined by the curve's tangent and curvature. The Frenet frame changes orientation
over the length of the curve. It is defined as normalized orthogonal vectors with w in the direction of the
first derivative ( P 00 ( s )),

and u formed to complete, for example, a left-handed coordinate system as computed in a right-handed
space ( Figure 3.31 ). Specifically, at a given parameter value s , the left-handed Frenet frame is calcu-
lated according to Equation 3.32, as illustrated in Figure 3.32 . The vectors are then normalized.

P 0 ( s )), v orthogonal to w and in the general direction of the second derivative (

P 0 w ¼ ð s Þ

P 0 P 00

(3.32)

u ¼ ð s Þ ð s Þ

v ¼ u w

P ( s )

P ( s )

u

P ( s )

P ( s )

FIGURE 3.31
The derivatives at a point along the curve are used to form the u vector.

v

w

u

FIGURE 3.32
Left-handed Frenet frame at a point along a curve.

4 Note the potential source of confusion between the use of the term frame to mean (1) a frame of animation and (2) the
moving coordinate system of the Frenet frame. The context in which the term frame is used should determine its meaning.