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 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.

The quadratic surfaces of revolution:

• 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 $a < b$ , 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 $$\frac{x^2}{a^2}-\frac{y^2+z^2}{b^2} = 1,$$ the one-sheeted hyperboloid $$\frac{x^2}{a^2}-\frac{y^2+z^2}{b^2} = -1$$ and the cone of revolution $$\frac{x^2}{a^2}-\frac{y^2+z^2}{b^2} = 0,$$ which apparently is the common asymptote cone of both hyperboloids.

Bibliography

1

LAURI PIMIÄ: Analyyttinen geometria. Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1958).