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SURFACE OF REVOLUTION


Cylindrical equation of the surfaces of revolution with axis Oz.
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 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 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 (H1 and H2) 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