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\begin{document}

 If a \htmladdnormallink{curve}{http://planetmath.org/encyclopedia/LocalMultiplicity.html} in $\mathbb{R}^3$ \htmladdnormallink{rotates}{http://planetmath.org/encyclopedia/Rotate.html} about a \htmladdnormallink{line}{http://planetmath.org/encyclopedia/Incident2.html}, it \htmladdnormallink{generates}{http://planetmath.org/encyclopedia/SubgroupGeneratedBy.html} a {\em surface of revolution}. The line is called the {\em axis of revolution}. Every \htmladdnormallink{point}{http://planetmath.org/encyclopedia/Point.html} of the curve generates a {\em circle of latitude}. If the \htmladdnormallink{surface}{http://planetmath.org/encyclopedia/Surface.html} is intersected by a half-plane beginning from the axis of revolution, the \htmladdnormallink{intersection}{http://planetmath.org/encyclopedia/Meets.html} curve is a {\em meridian curve}. One can always think that the surface of revolution is \htmladdnormallink{generated by}{http://planetmath.org/encyclopedia/SubgroupGeneratedBy.html} the \htmladdnormallink{rotation}{http://planetmath.org/encyclopedia/Rotate.html} of a certain meridian, which may be called the {\em 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, \htmladdnormallink{parallel}{http://planetmath.org/encyclopedia/ParallelLines.html} to the $yz$-plane, with \htmladdnormallink{centre}{http://planetmath.org/encyclopedia/Centre4.html} on the $x$-axis and with the \htmladdnormallink{radius}{http://planetmath.org/encyclopedia/NSphere2.html} $|f(x)|$. So the $y$- and $z$-coordinates of each point on this \htmladdnormallink{circle}{http://planetmath.org/encyclopedia/InteriorPoint2.html} satisfy the \htmladdnormallink{equation}{http://planetmath.org/encyclopedia/MultipleRoot.html}
$$y^2+z^2 = [f(x)]^2.$$
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
$$F(x,\,\sqrt{y^2\!+\!z^2}) = 0.$$

\textbf{Examples.}

When the \htmladdnormallink{catenary}{http://planetmath.org/encyclopedia/Catenary2.html} \,$y = a\cosh\frac{x}{a}$\, rotates about the $x$-axis, it generates the {\em catenoid}
$$y^2+z^2 = a^2\cosh^2\frac{x}{a}.$$
The catenoid is the only surface of revolution being also a \htmladdnormallink{minimal surface}{http://planetmath.org/encyclopedia/MinimalSurface.html}.

The \htmladdnormallink{quadratic surfaces}{http://planetmath.org/encyclopedia/ConeSurface.html} of revolution:
\begin{itemize}
\item When the \htmladdnormallink{ellipse}{http://planetmath.org/encyclopedia/DandelinSphere.html} \,$\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$\, rotates about the $x$-axis, we get the \htmladdnormallink{ellipsoid}{http://planetmath.org/encyclopedia/Ellipsoid.html}
$$\frac{x^2}{a^2}+\frac{y^2+z^2}{b^2} = 1.$$
This is a {\em stretched ellipsoid}, if\, $a > b$,\, and a {\em flattened ellipsoid}, if\, $a < b$, and a \htmladdnormallink{sphere}{http://planetmath.org/encyclopedia/NSphere2.html} of radius $a$, if\, $a = b$.
\item When the \htmladdnormallink{parabola}{http://planetmath.org/encyclopedia/Focus4.html} \,$y^2 = 2px$ (with $p$ the {\em \htmladdnormallink{latus rectum}{http://planetmath.org/encyclopedia/DandelinSphere.html}} or the \htmladdnormallink{parameter of parabola}{http://planetmath.org/encyclopedia/ParametricPresentation.html}) rotates about the $x$-axis, we get the {\em paraboloid of revolution}
$$y^2+z^2 = 2px.$$
\item When we let the \htmladdnormallink{conjugate hyperbolas}{http://planetmath.org/encyclopedia/ConjugateHyperbola.html} and their common \htmladdnormallink{asymptotes}{http://planetmath.org/encyclopedia/Asymptote.html}
\,$\displaystyle\frac{x^2}{a^2}-\frac{y^2}{b^2} = s$\, (with\, $s = 1,\,-1,\,0$) rotate about the $x$-axis, we obtain the {\em \htmladdnormallink{two-sheeted hyperboloid}{http://planetmath.org/encyclopedia/ConeSurface.html}}
$$\frac{x^2}{a^2}-\frac{y^2+z^2}{b^2} = 1,$$
the {\em \htmladdnormallink{one-sheeted hyperboloid}{http://planetmath.org/encyclopedia/ConeSurface.html}}
$$\frac{x^2}{a^2}-\frac{y^2+z^2}{b^2} = -1$$
and the {\em cone of revolution}
$$\frac{x^2}{a^2}-\frac{y^2+z^2}{b^2} = 0,$$
which apparently is the common {\em asymptote cone} of both hyperboloids.
\end{itemize}



\begin{thebibliography}{8}
\bibitem{LP}{\sc Lauri Pimi\"a}: {\em Analyyttinen geometria}.\, Werner S\"oderstr\"om Osakeyhti\"o, Porvoo and Helsinki (1958).
\end{thebibliography}


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