Fig. 3.

Description of body postures and trails of C. elegans in 3D. (A) Definition of the trail coordinate $s$ and the worm coordinate $s\prime$. (B) Rotation of the Serret–Frenet basis $(\widehat{\text{𝒕}},\widehat{\text{𝒏}},\widehat{\text{𝒃}})$ as it moves along the curve. The curvature $\kappa$ is associated with rotation around the binormal unit vector $\widehat{\text{𝒃}}$; deformations resulting from nonzero curvature thus occur in the $\widehat{\text{𝒕}}$–$\widehat{\text{𝒏}}$ plane. Torsion $\tau$ is associated with rotation around the tangent unit vector $\widehat{\text{𝒕}}$; a nonzero torsion results in out-of-plane curve deformation, because the local deformation plane $(\widehat{\text{𝒕}},\widehat{\text{𝒏}})$ changes its orientation.