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Natural coordinate frame

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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)
How to Cite This Entry:
Natural coordinate frame. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Natural_coordinate_frame&oldid=14256
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article