Difference between revisions of "Natural coordinate frame"
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''Frénet trihedron, Frénet frame, natural trihedron'' | ''Frénet trihedron, Frénet frame, natural trihedron'' | ||
A figure consisting of the tangent, the [[Principal normal|principal normal]] and the [[Binormal|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|Natural parameter]]) is, in a neighbourhood of that point, | A figure consisting of the tangent, the [[Principal normal|principal normal]] and the [[Binormal|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|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|curvature]] and [[Torsion|torsion]] of the curve at the point. | ||
====Comments==== | ====Comments==== | ||
Latest revision as of 08:02, 6 June 2020
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=47948
Natural coordinate frame. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Natural_coordinate_frame&oldid=47948
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article