Rotation surface

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
[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)