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Difference between revisions of "Rotation indicatrix"

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''rotation diagram''
 
''rotation diagram''
  
One of the twelve [[Darboux surfaces|Darboux surfaces]] associated with the [[Infinitesimal deformation|infinitesimal deformation]] of a surface. It is the set of points in space described by a position vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082630/r0826301.png" /> which are parallel to the rotation vector (instantaneous angular velocity) defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082630/r0826302.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082630/r0826303.png" /> is the velocity vector of the infinitesimal deformation of the surface described by the position vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082630/r0826304.png" />. The displacement indicatrix (displacement diagram) is defined in a similar manner by the displacement vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082630/r0826305.png" />.
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One of the twelve [[Darboux surfaces|Darboux surfaces]] associated with the [[Infinitesimal deformation|infinitesimal deformation]] of a surface. It is the set of points in space described by a position vector $  \mathbf y $
 +
which are parallel to the rotation vector (instantaneous angular velocity) defined by the equation $  d \mathbf z = [ \mathbf y  d \mathbf x ] $,  
 +
where $  \mathbf z $
 +
is the velocity vector of the infinitesimal deformation of the surface described by the position vector $  \mathbf x $.  
 +
The displacement indicatrix (displacement diagram) is defined in a similar manner by the displacement vector $  \mathbf s = \mathbf z - [ \mathbf y \mathbf x ] $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.V. Efimov,  "Qualitative questions of the theory of deformations of surfaces"  ''Uspekhi Mat. Nauk'' , '''3''' :  2  (1948)  pp. 47–158  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.E. Cohn-Vossen,  "Some problems of differential geometry in the large" , Moscow  (1959)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.V. Efimov,  "Qualitative questions of the theory of deformations of surfaces"  ''Uspekhi Mat. Nauk'' , '''3''' :  2  (1948)  pp. 47–158  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.E. Cohn-Vossen,  "Some problems of differential geometry in the large" , Moscow  (1959)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Darboux,  "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1–4''' , Chelsea, reprint  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  pp. 1–5</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.V. Efimov,  "Qualitative problems of the theory of deformation of surfaces" , Amer. Math. Soc.  (1951)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Darboux,  "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1–4''' , Chelsea, reprint  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  pp. 1–5</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.V. Efimov,  "Qualitative problems of the theory of deformation of surfaces" , Amer. Math. Soc.  (1951)  (Translated from Russian)</TD></TR></table>

Latest revision as of 08:12, 6 June 2020


rotation diagram

One of the twelve Darboux surfaces associated with the infinitesimal deformation of a surface. It is the set of points in space described by a position vector $ \mathbf y $ which are parallel to the rotation vector (instantaneous angular velocity) defined by the equation $ d \mathbf z = [ \mathbf y d \mathbf x ] $, where $ \mathbf z $ is the velocity vector of the infinitesimal deformation of the surface described by the position vector $ \mathbf x $. The displacement indicatrix (displacement diagram) is defined in a similar manner by the displacement vector $ \mathbf s = \mathbf z - [ \mathbf y \mathbf x ] $.

References

[1] N.V. Efimov, "Qualitative questions of the theory of deformations of surfaces" Uspekhi Mat. Nauk , 3 : 2 (1948) pp. 47–158 (In Russian)
[2] S.E. Cohn-Vossen, "Some problems of differential geometry in the large" , Moscow (1959) (In Russian)

Comments

References

[a1] G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1–4 , Chelsea, reprint (1972)
[a2] M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish pp. 1–5
[a3] N.V. Efimov, "Qualitative problems of the theory of deformation of surfaces" , Amer. Math. Soc. (1951) (Translated from Russian)
How to Cite This Entry:
Rotation indicatrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rotation_indicatrix&oldid=48588
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article