Frénet trihedron

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natural trihedron

The trihedral angle formed by the rays emanating from a point of a regular curve in the respective directions of the tangent , the normal and the binormal to the curve. If the coordinate axes, respectively, lie along the sides of the Frénet trihedron, then the equation of the curve in this coordinate system has the form

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

Figure: f041700a

Figure: f041700b

Figure: f041700c

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



[a1] C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4
How to Cite This Entry:
Frénet trihedron. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article