Frénet trihedron

natural trihedron

The trihedral angle formed by the rays emanating from a point $P$ of a regular curve $\gamma$ in the respective directions of the tangent $\tau$, the normal $\nu$ and the binormal $\beta$ to the curve. If the $x,y,z$ coordinate axes, respectively, lie along the sides of the Frénet trihedron, then the equation of the curve in this coordinate system has the form

$$x=\Delta s-\frac{k_1^2\Delta s^3}{6}+o(\Delta s^3),$$

$$y=\frac{k_1\Delta s^2}{2}+\frac{k_1'\Delta s^3}{6}+o(\Delta s^3),$$

$$z=-\frac{k_1k_2}{6}\Delta s^3+o(\Delta s^3),$$

where $k_1$ and $k_2$ are the curvature and torsion of the curve, and $s$ is the natural parameter. The qualitative form of the projections of the curve onto the planes of the Frénet trihedron for $k_1\neq0$ and $k_2\neq0$ can be seen in the figures.

Figure: f041700a

Figure: f041700b

Figure: f041700c

This trihedron was studied by F. Frénet (1847).

References

[a1]	C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4