Natural coordinate frame

Frénet trihedron, Frénet frame, natural trihedron

A figure consisting of the tangent, the principal normal and the binormal of a space curve, and the three planes defined by the pairs of these straight lines. If the edges of the natural frame at a given point of a curve are taken as the axes of a Cartesian coordinate system, the equation of the curve in the natural parametrization (see Natural parameter) is, in a neighbourhood of that point,

$$ x = s + \dots ,\ \ y = \frac{k _ {1} }{2 } s ^ {2} + \dots ,\ \ z = \frac{k _ {1} k _ {2} }{6 } s ^ {3} + \dots , $$

where $ k _ {1} $ and $ k _ {2} $ are the curvature and torsion of the curve at the point.

Comments

Cf. also Frénet trihedron.

References

[a1]	W. Klingenberg, "A course in differential geometry" , Springer (1978) (Translated from German)
[a2]	W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1977)