# Swerve of a curve

From Encyclopedia of Mathematics

*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 |

#### Comments

#### 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