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Homesurface of revolution
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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
$F(x,\,\sqrt{y^{2}\!+\!z^{2}})\;=\;0.$ 
Examples.
When the catenary $y=a\cosh\frac{x}{a}$ rotates about the $x$axis, it generates the catenoid
$y^{2}\!+\!z^{2}\;=\;a^{2}\cosh^{2}\frac{x}{a}.$ 
The catenoid is the only surface of revolution being also a minimal surface.
The quadratic surfaces of revolution:

When the 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
$y^{2}\!+\!z^{2}\;=\;2px.$ 
When we let the conjugate hyperbolas and their common asymptotes $\displaystyle\frac{x^{2}}{a^{2}}\frac{y^{2}}{b^{2}}=s$ (with $s=1,\,1,\,0$) rotate about the $x$axis, we obtain the twosheeted hyperboloid
$\frac{x^{2}}{a^{2}}\frac{y^{2}\!+\!z^{2}}{b^{2}}\;=\;1,$ the onesheeted hyperboloid
$\frac{x^{2}}{a^{2}}\frac{y^{2}\!+\!z^{2}}{b^{2}}\;=\;1$ and the cone of revolution
$\frac{x^{2}}{a^{2}}\frac{y^{2}\!+\!z^{2}}{b^{2}}\;=\;0,$ which apparently is the common asymptote cone of both hyperboloids.
References
 1 Lauri Pimiä: Analyyttinen geometria. Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1958).
Mathematics Subject Classification
57M20 no label found51M04 no label found Forums
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