# 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

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

The catenoid is the only surface of revolution being also a minimal surface.

• 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 stretched ellipsoid, if  $a>b$,  and a flattened ellipsoid, if  $a, 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

 $y^{2}\!+\!z^{2}\;=\;2px.$
• 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 two-sheeted hyperboloid

 $\frac{x^{2}}{a^{2}}-\frac{y^{2}\!+\!z^{2}}{b^{2}}\;=\;1,$
 $\frac{x^{2}}{a^{2}}-\frac{y^{2}\!+\!z^{2}}{b^{2}}\;=\;-1$

and the cone of revolution

 $\frac{x^{2}}{a^{2}}-\frac{y^{2}\!+\!z^{2}}{b^{2}}\;=\;0,$

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 2013-03-22 17:17:08 Last modified on 2013-03-22 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 0-meridian Defines cone of revolution Defines asymptote cone Defines catenoid