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''self-rotation of a curve''
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091530/s0915301.png" />, the swerve is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091530/s0915302.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091530/s0915303.png" /> are the variations under right and left traversal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091530/s0915304.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091530/s0915305.png" /> is the variation of the curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091530/s0915306.png" /> as a set. Curves with swerve zero are called quasi-geodesic curves (cf. [[Quasi-geodesic line|Quasi-geodesic line]]).
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====References====
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''self-rotation of a curve''
<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>
 
 
 
 
 
 
 
====Comments====
 
  
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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 $,
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the swerve is equal to  $  ( \sigma _ {r} + \sigma _ {l} - \Omega )/2 $,
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where  $  \sigma _ {r} , \sigma _ {l} $
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are the variations under right and left traversal of  $  L $,
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while  $  \Omega $
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is the variation of the curvature of  $  L $
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as a set. Curves with swerve zero are called quasi-geodesic curves (cf. [[Quasi-geodesic line]]).
  
 
====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>
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<table>
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<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>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Busemann,  "Convex surfaces" , Interscience  (1958)  pp. Chapt. III, Sect. 15</TD></TR>
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</table>

Latest revision as of 05:43, 9 April 2023


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
[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=15805
This article was adapted from an original article by Yu.D. Burago (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article