To get clear on what’s meant by movement (2), seize your thumb with you left hand and make as should you were attempting to unscrew it from your hand. This suggests a simple option to make a rocker. This observation suggests a easy approach to express the distinction between flies and birds-flies fly with a lot larger curvature and torsion than do the birds. Some preliminary issues to notice are that if H is much smaller than R, you get a curvature roughly equal to 1/R, just like for a circle, and a tau very close to 0. If, alternatively, R may be very near zero, then the torsion is roughly 1/H whereas the curvature is close to 0. A fly which does a barrel-roll while transferring by a practically straight distance of H has a torsion of 1/H. The sooner it may possibly roll, the higher is its torsion. This can be a sort of “yawing” motion, and it corresponds to the primary of the three Serret-Frenet formulae: the change in the tangent is equal to the curvature times regular. The first and third equations correspond, respectively, to the definitions of curvature and torsion. I feel that they generally transfer alongside at a constant speed like a space curve parameterized by its arclength, and that they handle to loiter here and pace away from there by various their curvature and torsion between low and excessive values
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