# Difference between revisions of "Rotation surface"

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+ | $#C+1 = 19 : ~/encyclopedia/old_files/data/R082/R.0802670 Rotation surface, | ||

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''surface of rotation, rotational surface'' | ''surface of rotation, rotational surface'' | ||

− | A surface generated by the rotation of a plane curve | + | 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|Gaussian curvature]] is | + | The [[Gaussian curvature|Gaussian curvature]] is $ K = - z ^ \prime M/ \rho N ^ {4} $, |

+ | the [[Mean curvature|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|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 | ||

− | + | $$ \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 | + | 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