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