# Swerve of a curve

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self-rotation of a curve

The part of the variation of the rotation of a curve on an irregular surface not caused by the concentration of the integral curvature of the surface on the set of points of the curve. For a simple arc $L$, the swerve is equal to $( \sigma _ {r} + \sigma _ {l} - \Omega )/2$, where $\sigma _ {r} , \sigma _ {l}$ are the variations under right and left traversal of $L$, while $\Omega$ is the variation of the curvature of $L$ as a set. Curves with swerve zero are called quasi-geodesic curves (cf. Quasi-geodesic line).

#### References

 [1] A.D. Aleksandrov, V.V. Strel'tsov, "Isoperimetric problems and estimates of the length of a curve on a surface" Proc. Steklov Inst. Math. , 76 (1965) pp. 81–99 Trudy Mat. Inst. Steklov. , 76 (1965) pp. 67–80

#### References

 [a1] H. Busemann, "Convex surfaces" , Interscience (1958) pp. Chapt. III, Sect. 15
How to Cite This Entry:
Swerve of a curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Swerve_of_a_curve&oldid=48918
This article was adapted from an original article by Yu.D. Burago (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article