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SURFACE OF REVOLUTION
Note: the following functions denoted by f are
not identical!
General equation of the surfaces of revolution with axis directed by (a, b, c) : . General equation of the surfaces of revolution with axis Oz: . Cylindrical equation: . 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. Gaussian 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 volume of the solid of revolution generated
by the rotation of a planar domain around an axis of its plane that does
not cross the domain is equal to
where S is the area of the domain and d the distance from
the center of gravity of the domain to the axis.

A surface of revolution (or rotation surface)
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 halfplanes
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 Tannery's
pear
 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 
line  secant line  cone of revolution 
line  parallel line  cylinder of revolution 
line  non coplanar line  onesheeted 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  onesheeted hyperboloid of revolution 
hyperbola  focal axis  twosheeted 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. 

Cubic surface of revolution with equation ; its meridian curve is studied on the page of the witch of Agnesi. 
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© Robert FERRÉOL Alain ESCULIER 2022