Most biological motions are three-dimensional. This includes the trajectories of whole organisms and of their appendages. While recordings of three-dimensional trajectories are sometimes published, quantitative analysis of these trajectories is uncommon, primarily because there are no standard techniques or conventions in biology for the analysis of three-dimensional trajectories. This paper describes a new technique, finite helix fit (FHF), based on the geometry of three-dimensional curves, whereby a three-dimensional trajectory is completely described by its velocity, curvature and torsion. FHF estimates these parameters from discretely sampled points on a trajectory (i.e. from positional data such as x,y,z coordinates). Other measures of motion can be derived from these parameters, such as the translational and rotational (or angular) velocities of an organism. The performance of the algorithms is demonstrated using simulated trajectories and trajectories of freely swimming organisms (a flagellate, Chlamydomonas reinhardtii; a ciliate, Paramecium tetraurelia; spermatozoa of a sea urchin, Arbacia punctulata; larvae of an ascidian, Botrylloides sp.).

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