surface of revolution

If a curve in ${ℝ}^3$ 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 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

$$F(x,\sqrt{y^2+z^2})=\mathrm{\hspace{0.33em}0}.$$

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 ellipse  ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$  rotates about the $x$-axis, we get the ellipsoid

  $$\frac{x^2}{a^2}+\frac{y^2+z^2}{b^2}=\mathrm{\hspace{0.33em}1}.$$

  This is a stretched ellipsoid, if  $a>b$,  and a flattened ellipsoid, if  , 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

  $$y^2+z^2=\mathrm{\hspace{0.33em}2}px.$$

• When we let the conjugate hyperbolas and their common asymptotes ${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=s$  (with  $s=1,-1,\mathrm{\hspace{0.17em}0}$) rotate about the $x$-axis, we obtain the two-sheeted hyperboloid

  $$\frac{x^2}{a^2}-\frac{y^2+z^2}{b^2}=\mathrm{\hspace{0.33em}1},$$

  the one-sheeted hyperboloid

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

  and the cone of revolution

  $$\frac{x^2}{a^2}-\frac{y^2+z^2}{b^2}=\mathrm{\hspace{0.33em}0},$$

  which apparently is the common asymptote cone of both hyperboloids.