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''surface of rotation, rotational surface''
 
''surface of rotation, rotational surface''
  
A surface generated by the rotation of a plane curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r0826701.png" /> around an axis in its plane. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r0826702.png" /> is defined by the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r0826703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r0826704.png" />, the position vector of the surface of rotation is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r0826705.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r0826706.png" /> is the parameter of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r0826707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r0826708.png" /> is the distance between a point on the surface and the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r0826709.png" /> of rotation and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r08267010.png" /> is the angle of rotation. The line element of the surface of rotation is
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A surface generated by the rotation of a plane curve $  L $
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around an axis in its plane. If $  L $
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is defined by the equations $  \rho = \rho ( u) $,  
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$  z = z( u) $,  
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the position vector of the surface of rotation is $  \mathbf r = \{ \rho ( u)  \cos  v, \rho ( u)  \sin  v, z( u) \} $,  
 +
where $  u $
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is the parameter of the curve $  L $,  
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$  \rho $
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is the distance between a point on the surface and the axis $  z $
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of rotation and $  v $
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is the angle of rotation. The line element of the surface of rotation is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r08267011.png" /></td> </tr></table>
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$$
 +
ds  ^ {2}  = \
 +
( \rho ^ {\prime 2 } + z ^ {\prime 2 } ) \
 +
du  ^ {2} + \rho  ^ {2}  dv  ^ {2} .
 +
$$
  
The [[Gaussian curvature|Gaussian curvature]] is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r08267012.png" />, the [[Mean curvature|mean curvature]] is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r08267013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r08267014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r08267015.png" />. The lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r08267016.png" /> are called parallels of the surface of rotation and are circles located in a plane normal to the axis of rotation, with their centres on this axis. The lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r08267017.png" /> are called meridians; they are all congruent to the rotating curve and lie in planes passing through the axis of rotation. The meridians and the parallels of a surface of rotation are its curvature lines and form an [[Isothermal net|isothermal net]].
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The [[Gaussian curvature|Gaussian curvature]] is $  K = - z  ^  \prime  M/ \rho N  ^ {4} $,  
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the [[Mean curvature|mean curvature]] is $  H = ( z  ^  \prime  N  ^ {2} - \rho M)/ 2 \rho N  ^ {3} $,  
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where $  M = z  ^  \prime  \rho  ^ {\prime\prime} - z  ^ {\prime\prime} \rho  ^  \prime  $,  
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$  N = \sqrt {\rho ^ {\prime 2 } + z ^ {\prime 2 } } $.  
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The lines $  u = \textrm{ const } $
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are called parallels of the surface of rotation and are circles located in a plane normal to the axis of rotation, with their centres on this axis. The lines $  v = \textrm{ const } $
 +
are called meridians; they are all congruent to the rotating curve and lie in planes passing through the axis of rotation. The meridians and the parallels of a surface of rotation are its curvature lines and form an [[Isothermal net|isothermal net]].
  
 
A surface of rotation allows for a [[Deformation|deformation]] into another surface of rotation, under which its net of curvature lines is preserved and therefore is a principal base of the deformation. The umbilical points (cf. [[Umbilical point|Umbilical point]]) of a surface of rotation are characterized by the property that the centre of curvature of the meridian lies on the axis of rotation. The product of the radius of a parallel by the cosine of the angle of intersection of the surface of rotation with the parallel is constant along a geodesic (Clairaut's theorem).
 
A surface of rotation allows for a [[Deformation|deformation]] into another surface of rotation, under which its net of curvature lines is preserved and therefore is a principal base of the deformation. The umbilical points (cf. [[Umbilical point|Umbilical point]]) of a surface of rotation are characterized by the property that the centre of curvature of the meridian lies on the axis of rotation. The product of the radius of a parallel by the cosine of the angle of intersection of the surface of rotation with the parallel is constant along a geodesic (Clairaut's theorem).
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The metric of a surface of rotation can be presented in the form
 
The metric of a surface of rotation can be presented in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r08267018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$ \tag{1 }
 +
ds  ^ {2}  = \Lambda  ^ {2} ( r)
 +
( dx  ^ {2} + dy  ^ {2} ) ,\ \
 +
r  ^ {2} = x  ^ {2} + y  ^ {2} .
 +
$$
  
For the existence of metrics of the form (1) and for isometric immersions of these in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082670/r08267019.png" /> as surfaces of rotation see [[#References|[1]]].
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For the existence of metrics of the form (1) and for isometric immersions of these in $  \mathbf R  ^ {n} $
 +
as surfaces of rotation see [[#References|[1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.Kh. Sabitov,  , ''Abstracts Coll. Diff. Geom. (August 1989, Eger, Hungary)''  pp. 47–48</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.Kh. Sabitov,  , ''Abstracts Coll. Diff. Geom. (August 1989, Eger, Hungary)''  pp. 47–48</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.P. Do Carmo,  "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)  pp. 145</TD></TR><TR><TD valign="top">[a3]</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">[a4]</TD> <TD valign="top">  K. Leichtweiss,  "Einführung in die Differentialgeometrie" , Springer  (1973)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.P. Do Carmo,  "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)  pp. 145</TD></TR><TR><TD valign="top">[a3]</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">[a4]</TD> <TD valign="top">  K. Leichtweiss,  "Einführung in die Differentialgeometrie" , Springer  (1973)</TD></TR></table>

Latest revision as of 08:12, 6 June 2020


surface of rotation, rotational surface

A surface generated by the rotation of a plane curve $ L $ around an axis in its plane. If $ L $ is defined by the equations $ \rho = \rho ( u) $, $ z = z( u) $, the position vector of the surface of rotation is $ \mathbf r = \{ \rho ( u) \cos v, \rho ( u) \sin v, z( u) \} $, where $ u $ is the parameter of the curve $ L $, $ \rho $ is the distance between a point on the surface and the axis $ z $ of rotation and $ v $ is the angle of rotation. The line element of the surface of rotation is

$$ ds ^ {2} = \ ( \rho ^ {\prime 2 } + z ^ {\prime 2 } ) \ du ^ {2} + \rho ^ {2} dv ^ {2} . $$

The Gaussian curvature is $ K = - z ^ \prime M/ \rho N ^ {4} $, the mean curvature is $ H = ( z ^ \prime N ^ {2} - \rho M)/ 2 \rho N ^ {3} $, where $ M = z ^ \prime \rho ^ {\prime\prime} - z ^ {\prime\prime} \rho ^ \prime $, $ N = \sqrt {\rho ^ {\prime 2 } + z ^ {\prime 2 } } $. The lines $ u = \textrm{ const } $ are called parallels of the surface of rotation and are circles located in a plane normal to the axis of rotation, with their centres on this axis. The lines $ v = \textrm{ const } $ are called meridians; they are all congruent to the rotating curve and lie in planes passing through the axis of rotation. The meridians and the parallels of a surface of rotation are its curvature lines and form an isothermal net.

A surface of rotation allows for a deformation into another surface of rotation, under which its net of curvature lines is preserved and therefore is a principal base of the deformation. The umbilical points (cf. Umbilical point) of a surface of rotation are characterized by the property that the centre of curvature of the meridian lies on the axis of rotation. The product of the radius of a parallel by the cosine of the angle of intersection of the surface of rotation with the parallel is constant along a geodesic (Clairaut's theorem).

The only minimal surface of rotation is the catenoid. A ruled surface of rotation is a one-sheet hyperboloid or one of its degeneracies: a cylinder, a cone or a plane. A surface of rotation with more than one axis of rotation is a sphere or a plane.

The metric of a surface of rotation can be presented in the form

$$ \tag{1 } ds ^ {2} = \Lambda ^ {2} ( r) ( dx ^ {2} + dy ^ {2} ) ,\ \ r ^ {2} = x ^ {2} + y ^ {2} . $$

For the existence of metrics of the form (1) and for isometric immersions of these in $ \mathbf R ^ {n} $ as surfaces of rotation see [1].

References

[1] I.Kh. Sabitov, , Abstracts Coll. Diff. Geom. (August 1989, Eger, Hungary) pp. 47–48

Comments

References

[a1] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2] M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 145
[a3] M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish pp. 1–5
[a4] K. Leichtweiss, "Einführung in die Differentialgeometrie" , Springer (1973)
How to Cite This Entry:
Rotation surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rotation_surface&oldid=18444
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article