PlanetMath: surface of revolution

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surface of revolution

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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).

"surface of revolution" is owned by pahio.

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Also defines:

surface of revolution, axis of revolution, circle of latitude, meridian curve, 0-meridian, cone of revolution, asymptote cone, catenoid

Keywords:

rotation

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Attachments:: Torricelli's trumpet (Definition) by pahio

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Cross-references: one-sheeted hyperboloid, two-sheeted hyperboloid, asymptotes, conjugate hyperbolas, parameter of parabola, latus rectum, parabola, sphere, ellipsoid, ellipse, quadratic surfaces, minimal surface, catenary, equation, circle, radius, centre, parallel, rotation, generated by, intersection, surface, point, generates, line, rotates, curve
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This is version 8 of surface of revolution, born on 2007-06-20, modified 2007-10-16.
Object id is 9627, canonical name is SurfaceOfRevolution2.
Accessed 2374 times total.

Classification:

AMS MSC:	51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)
	57M20 (Manifolds and cell complexes :: Low-dimensional topology :: Two-dimensional complexes)

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