# Rotation surface

*surface of rotation, rotational surface*

A surface generated by the rotation of a plane curve around an axis in its plane. If is defined by the equations , , the position vector of the surface of rotation is , where is the parameter of the curve , is the distance between a point on the surface and the axis of rotation and is the angle of rotation. The line element of the surface of rotation is

The Gaussian curvature is , the mean curvature is , where , . The lines 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 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

(1) |

For the existence of metrics of the form (1) and for isometric immersions of these in 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