surface of revolution
If a curve in ${\mathbb{R}}^{3}$ rotates about a line, it generates a surface of revolution^{}. 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 halfplane 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 meridian^{}, which may be called the 0meridian.
Let $y=f(x)$ be a curve of the $xy$plane rotating about the $x$axis. Then any point $(x,y)$ of this 0meridian draws a circle of latitude, parallel^{} 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
$${y}^{2}+{z}^{2}={[f(x)]}^{2}.$$ 
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
$$F(x,\sqrt{{y}^{2}+{z}^{2}})=\mathrm{\hspace{0.33em}0}.$$ 
Examples.
When the catenary $y=a\mathrm{cosh}\frac{x}{a}$ rotates about the $x$axis, it generates the catenoid
$${y}^{2}+{z}^{2}={a}^{2}{\mathrm{cosh}}^{2}\frac{x}{a}.$$ 
The catenoid is the only surface of revolution being also a minimal surface^{}.
The quadratic surfaces of revolution:

•
When the ellipse^{} $\frac{{x}^{2}}{{a}^{2}}}+{\displaystyle \frac{{y}^{2}}{{b}^{2}}}=1$ rotates about the $x$axis, we get the ellipsoid^{}
$$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}+{z}^{2}}{{b}^{2}}=\mathrm{\hspace{0.33em}1}.$$ This is a stretched ellipsoid, if $a>b$, and a flattened ellipsoid, if $$, and a sphere of radius $a$, if $a=b$.

•
When the parabola ${y}^{2}=2px$ (with $p$ the latus rectum or the parameter of parabola) rotates about the $x$axis, we get the paraboloid of revolution
$${y}^{2}+{z}^{2}=\mathrm{\hspace{0.33em}2}px.$$ 
•
When we let the conjugate hyperbolas and their common asymptotes $\frac{{x}^{2}}{{a}^{2}}}{\displaystyle \frac{{y}^{2}}{{b}^{2}}}=s$ (with $s=1,1,\mathrm{\hspace{0.17em}0}$) rotate about the $x$axis, we obtain the twosheeted hyperboloid
$$\frac{{x}^{2}}{{a}^{2}}\frac{{y}^{2}+{z}^{2}}{{b}^{2}}=\mathrm{\hspace{0.33em}1},$$ $$\frac{{x}^{2}}{{a}^{2}}\frac{{y}^{2}+{z}^{2}}{{b}^{2}}=1$$ and the cone of revolution
$$\frac{{x}^{2}}{{a}^{2}}\frac{{y}^{2}+{z}^{2}}{{b}^{2}}=\mathrm{\hspace{0.33em}0},$$ which apparently is the common asymptote cone of both hyperboloids.
References
 1 Lauri Pimiä: Analyyttinen geometria. Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1958).
Title  surface of revolution 
Canonical name  SurfaceOfRevolution 
Date of creation  20130322 17:17:08 
Last modified on  20130322 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  0meridian 
Defines  cone of revolution 
Defines  asymptote cone 
Defines  catenoid 