Difference between revisions of "Swerve of a curve"
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''self-rotation of a curve'' | ''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 | + | 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|Quasi-geodesic line]]). | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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</TD></TR></table> | ||
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====Comments==== | ====Comments==== | ||
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====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Busemann, "Convex surfaces" , Interscience (1958) pp. Chapt. III, Sect. 15</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Busemann, "Convex surfaces" , Interscience (1958) pp. Chapt. III, Sect. 15</TD></TR></table> | ||
Revision as of 08:24, 6 June 2020
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
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