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Animating the Pectoral Flapping Motion

To achieve the flapping motion, let us define displacement vectors tex2html_wrap_inline4064 which, when added to tex2html_wrap_inline4066 , i=1,2,3,4, yield the new directions of tex2html_wrap_inline4066 . Let the normal of the plane formed by tex2html_wrap_inline4072 and tex2html_wrap_inline4074 be a unit vector tex2html_wrap_inline4076 . The pectoral flapping motions are modeled by simply specifying the direction of the displacement vectors tex2html_wrap_inline4064 along tex2html_wrap_inline4076 and then letting the lengths of tex2html_wrap_inline4064 vary over time in a sinusoidal fashion. In particular, the tex2html_wrap_inline4064 's are defined as follows:

eqnarray2225

where a and b are parameters that define the amplitude and frequency of flapping, respectively; time t is discrete and equals the number of animation frames. Since the tex2html_wrap_inline4066 's are not coplanar (they are nearly coplanar), the shape of the fins deform slightly during the flapping motion. This is not undesirable since natural pectoral fins deform constantly due to hydrodynamic forces.

In our implementation, we choose a and b to be nearly proportional to tex2html_wrap_inline4098 such that the faster the fish needs to ascend or descend, the greater the amplitude and frequency with which the fins flap. Note that a and b are nonzero values when tex2html_wrap_inline4104 so the fins are kept in motion even when the fish is not engaged in pitching, yawing, or rolling. Fig. gif shows four snapshots of the flapping motion of the pectoral fins.


next up previous contents
Next: Animating the Pectoral Oaring Motion Up: Visualization of the Pectoral Motions Previous: Visualization of the Pectoral Motions
Xiaoyuan TuJanuary 1996