surface of revolution

If a curve in 3 rotates about a line, it generates a surface of revolutionMathworldPlanetmath. The line is called the axis of revolution.  Every point of the curve generates a circle of latitude. If the surface is intersected by a half-plane beginning from the axis of revolution, the intersection curve is a meridian curve. One can always think that the surface of revolution is generated by the rotation of a certain meridianMathworldPlanetmath, which may be called the 0-meridian.

Let  y=f(x)  be a curve of the xy-plane rotating about the x-axis. Then any point  (x,y)  of this 0-meridian draws a circle of latitude, parallelMathworldPlanetmathPlanetmath to the yz-plane, with centre on the x-axis and with the radius |f(x)|. So the y- and z-coordinates of each point on this circle satisfy the equation


This equation is thus satisfied by all points  (x,y,z)  of the surface of revolution and therefore it is the equation of the whole surface of revolution.

More generally, if the equation of the meridian curve in the xy-plane is given in the implicit form  F(x,y)=0,  then the equation of the surface of revolution may be written

F(x,y2+z2)= 0.


When the catenaryy=acoshxa  rotates about the x-axis, it generates the catenoid


The catenoid is the only surface of revolution being also a minimal surfaceMathworldPlanetmathPlanetmath.

The quadratic surfaces of revolution:

  • When the ellipsePlanetmathPlanetmathx2a2+y2b2=1  rotates about the x-axis, we get the ellipsoidMathworldPlanetmath

    x2a2+y2+z2b2= 1.

    This is a stretched ellipsoid, if  a>b,  and a flattened ellipsoid, if  a<b, and a sphere of radius a, if  a=b.

  • When the parabolay2=2px (with p the latus rectum or the parameter of parabola) rotates about the x-axis, we get the paraboloid of revolution

    y2+z2= 2px.
  • When we let the conjugate hyperbolas and their common asymptotesx2a2-y2b2=s  (with  s=1,-1, 0) rotate about the x-axis, we obtain the two-sheeted hyperboloid

    x2a2-y2+z2b2= 1,

    the one-sheeted hyperboloid


    and the cone of revolution

    x2a2-y2+z2b2= 0,

    which apparently is the common asymptote cone of both hyperboloids.


  • 1 Lauri Pimiä: Analyyttinen geometria.  Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1958).
Title surface of revolution
Canonical name SurfaceOfRevolution
Date of creation 2013-03-22 17:17:08
Last modified on 2013-03-22 17:17:08
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Topic
Classification msc 57M20
Classification msc 51M04
Related topic SurfaceOfRevolution
Related topic PappussTheoremForSurfacesOfRevolution
Related topic QuadraticSurfaces
Related topic ConicalSurface
Related topic Torus
Related topic SolidOfRevolution
Related topic LeastSurfaceOfRevolution
Related topic ConeInMathbbR3
Defines surface of revolution
Defines axis of revolution
Defines circle of latitude
Defines meridian curve
Defines 0-meridian
Defines cone of revolution
Defines asymptote cone
Defines catenoid