View source for Rotation surface

<!--
r0826701.png
$#A+1 = 19 n = 0
$#C+1 = 19 : ~/encyclopedia/old_files/data/R082/R.0802670 Rotation surface,
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

''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|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).

The only minimal surface of rotation is the [[Catenoid|catenoid]]. A ruled surface of rotation is a [[One-sheet hyperboloid|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 [[#References|[1]]].

====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>

====Comments====

====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>