# surface of revolution

If a curve in $\mathbb{R}^{3}$ rotates about a line, it generates a . 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 meridian  , 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, 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

 $y^{2}\!+\!z^{2}\;=\;a^{2}\cosh^{2}\frac{x}{a}.$