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Surface Of Revolution

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 be a curve of the -plane rotating about the -axis. Then any point $(x,\,y)$ of this 0-meridian draws a circle of latitude, parallel to the -plane, with centre on the -axis and with the radius $\vert f(x)\vert$ . So the - and -coordinates of each point on this circle satisfy the equation

$\displaystyle y^2+z^2 = [f(x)]^2.$

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 -plane is given in the implicit form $F(x,\,y) = 0$ , then the equation of the surface of revolution may be written

$\displaystyle F(x,\,\sqrt{y^2\!+\!z^2}) = 0.$

Examples.

When the catenary $y = a\cosh\frac{x}{a}$ rotates about the -axis, it generates the catenoid

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

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 -axis, we get the ellipsoid
$\displaystyle \frac{x^2}{a^2}+\frac{y^2+z^2}{b^2} = 1.$
This is a stretched ellipsoid, if , and a flattened ellipsoid, if , and a sphere of radius , if .
When the parabola (with the latus rectum or the parameter of parabola) rotates about the -axis, we get the paraboloid of revolution
$\displaystyle y^2+z^2 = 2px.$
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 -axis, we obtain the two-sheeted hyperboloid
$\displaystyle \frac{x^2}{a^2}-\frac{y^2+z^2}{b^2} = 1,$
the one-sheeted hyperboloid
$\displaystyle \frac{x^2}{a^2}-\frac{y^2+z^2}{b^2} = -1$
and the cone of revolution
$\displaystyle \frac{x^2}{a^2}-\frac{y^2+z^2}{b^2} = 0,$
which apparently is the common asymptote cone of both hyperboloids.

Bibliography

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

1	Curve
1	Incidence Geometry
1	Point
1	Euclidean Transformation
1	Parameter
1	Intersection
1	Generating Set Of A Group
1	Equation
2	Parallellism In Euclidean Plane
2	Sphere
2	Proof Of Uniqueness Of Center Of A Circle
2	Surface
3	Conic Section
4	Parabola
5	Midpoint
6	Ellipsoid
8	Asymptote
8	Quadratic Surfaces
15	Catenary
19	Conjugate Hyperbola
21	Plateaus Problem