Bulletin of Mathematical Biology

, Volume 55, Issue 1, pp 197–212 | Cite as

Orientation by helical motion—I. Kinematics of the helical motion of organisms with up to six degrees of freedom

  • Hugh C. Crenshaw


The kinematics of helical motion are descirbed for an organism treated as a rigid body with six degrees of freedom relative to the organism's frame of reference, i.e. the organism can translate in the direction of, or rotate around any of, three orthogonal axes fixed to its body. Equations are derived that express the unit vectors of the Frenet trihedron and the torsion and curvature of the trajectory in terms of the organism's translational and rotational velocities. These equations permit description of the radius, pitch, angular velocity and axis of a helical trajectory in terms of the translational and rotational velocities of the organism swimming along that trajectory. The results of this analysis are then used in two later papers that describe how organisms can orient to an external stimulus.


Rotational Velocity Unit Normal Vector Translational Velocity Unit Tangent Vector Helical Motion 
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Copyright information

© Society for Mathematical Biology 1992

Authors and Affiliations

  • Hugh C. Crenshaw
    • 1
  1. 1.Department of ZoologyDuke UniversityDurhamU.S.A.

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