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SURFACE OF REVOLUTION

Cylindrical equation of the surfaces of revolution with
axis Oz: .
Spherical equation: . Partial differential equation: . Cartesian parametrization: (generatrix ). In particular, for a plane generatrix : (). In the latter case: First fundamental quadratic form: . Surface element: . Second fundamental quadratic form: . Meridian curvature: ; parallel curvature: , where N is the normal
to the curve
with respect to the axis Oz.
Total curvature: ; mean curvature: . The curvature lines are the meridians and the parallels (). The cylindrical equation of the asymptotic lines is: . The cylindrical equation of the geodesics is: (see more at geodesics). Guldin theorems: The area of the surface of revolution generated
by the rotation of an arc of a plane curve around an axis of its plane
that does not cross the arc of the curve is equal to
where
l is the length of the arc of the curve and d the distance
from the center of gravity of the arc to the axis.
The |

A *surface of revolution* is a surface globally invariant
under the action of any rotation around a fixed line called *axis of
revolution*.

The rotation of a curve (called *generatrix*) around
a fixed line generates a surface of revolution.

The sections of a surface of revolution by half-planes
delimited by the axis of revolution, called *meridians*, are special
generatrices.

The sections by planes perpendicular to the axis are
circles called *parallels* of the surface (a surface of revolution
is therefore a circled surface).

The surfaces of revolution can also be defined as the tubes with variable section and linear bore, or as the envelopes of spheres the centers of which are aligned.

A surface is a portion of a surface of revolution iff the normal at every point meets, or is parallel to, a fixed line (which is the axis of revolution).

Examples:

- the plane

- the quadrics of revolution: cylinder
of revolution, cone
of revolution, sphere,
ellipsoid
of revolution, hyperboloids (*H*_{1}
and *H*_{2})
of revolution, paraboloid
of revolution

- the torus

- the catenoid,
the
onduloid, the nodoid

- the pseudosphere

- the revolution of the
sinusoid

- the hanging
drop of water

- Gabriel's
horn

- the tower
of constant pressure

- the surface of the solid
of maximal attraction

Here is a classification based on the nature of the generatrix:

Generatrix | Axis | Surface of revolution |

line | orthogonal line | plane |

secant line | cone of revolution | |

line | parallel line | cylinder of revolution |

line | non coplanar line | one-sheeted hyperboloid of revolution |

circle | diameter of the circle | sphere |

circle | in the plane of the circle | torus (the circular generatrices are the meridians) |

circle | meeting the axis of the circle | portion of a sphere |

circle | Oz |
torus (the circular generatrices are the Villarceau circles) |

ellipse | axis of the ellipse | ellipsoid of revolution |

parabola | axis of the parabola | paraboloid of revolution |

hyperbola | non focal axis | one-sheeted hyperboloid of revolution |

hyperbola | focal axis | two-sheeted hyperboloid of revolution |

rectangular hyperbola | asymptote | Gabriel's horn |

catenary | base | catenoid |

roulette of Delaunay | base | onduloid and nodoid |

tractrix | asymptote | pseudosphere |

logarithmic | asymptote | tower of constant pressure |

sinusoid | axis of translation | revolution of the sinusoid |

Animation showing the deformation of the surface of revolution generated by the rotation of a circle around an axis, starting from a torus (generated by its Villarceau circles), passing by the semisphere, and ending as a torus. |

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© Robert FERRÉOL Alain ESCULIER 2017