surface of revolutionSurfaceOfRevolution22007-06-20 06:07:152009-04-22 15:38:13Topicsurfacesurface of revolutionaxis of revolutioncircle of latitudemeridian curve0-meridiancone of revolutionasymptote conecatenoidrotation% this is the default PlanetMath preamble. as your knowledge
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If a curve in $\mathbb{R}^3$ rotates about a line, it generates a {\em surface of revolution}. The line is called the {\em axis of revolution}. Every point of the curve generates a {\em circle of latitude}. If the surface is intersected by a half-plane beginning from the axis of revolution, the intersection curve is a {\em 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 {\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, 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
$$F(x,\,\sqrt{y^2\!+\!z^2}) = 0.$$
\textbf{Examples.}
When the catenary \,$y = a\cosh\frac{x}{a}$\, 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 minimal surface.
The quadratic surfaces of revolution:
\begin{itemize}
\item When the ellipse \,$\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$\, rotates about the $x$-axis, we get the ellipsoid
$$\frac{x^2}{a^2}+\frac{y^2+z^2}{b^2} = 1.$$
This is a {\em stretched ellipsoid}, if\, $a > b$,\, and a {\em flattened ellipsoid}, if\, $a < b$, and a sphere of radius $a$, if\, $a = b$.
\item When the parabola \,$y^2 = 2px$ (with $p$ the {\em latus rectum} or the parameter of parabola) rotates about the $x$-axis, we get the {\em paraboloid of revolution}
$$y^2+z^2 = 2px.$$
\item 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 {\em two-sheeted hyperboloid}
$$\frac{x^2}{a^2}-\frac{y^2+z^2}{b^2} = 1,$$
the {\em one-sheeted hyperboloid}
$$\frac{x^2}{a^2}-\frac{y^2+z^2}{b^2} = -1$$
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}