# Surface of Revolution

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## surface of revolution

[¦sər·fəs əv ‚rev·ə′lü·shən]
(mathematics)
A surface realized by rotating a planar curve about some axis in its plane.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Surface of Revolution

a surface that can be generated by revolving a plane curve about a straight line, called the axis of the surface of revolution, lying in the plane of the curve. An example of such a surface is the sphere, which may be considered as the surface generated when a semicircle is revolved about its diameter. The curves formed by the intersection of a surface of revolution with planes passing through the axis are called meridians, and the curves of intersection of a surface of revolution with planes perpendicular to the axis are called parallels. If the z-axis of a rectangular system of coordinates x, y, and z is directed along the axis of a surface of revolution, then the parametric equations of the surface of revolution can be written

x = f(u] cos v y = f {u} sin v z = u

where f(u) is a function that determines the shape of the meridian and v is the angle of rotation of the plane of the meridian.

References in periodicals archive ?
Since a circle may be exactly represented by T-B-spline curve, we may find an exact representation of a surface of revolution provided we can represent r(v), z(v) in T-B-spline curve form.
The most convenient way to define a surface of revolution is to prescribe the (planar) generating curve, or generatrix, given by g(v) = [(r(v), 0, z(v)).sup.T] (take the curve in xoz plane as an example) and by the axis of revolution, in the same plane as
Then the T-B-spline surface with control points [P.sub.ij], knot vectors U and V, and shape parameter arrays [[lambda].sup.u] and [[lambda].sup.v] is a surface of revolution. Its generatrix is g(v), and its axis of revolution is z-axis.
Then the surface of revolution M with [x.sub.2]-axis may be given by
Tet M be a surface of revolution with timelike axis as (7).
Thus, there is no surface of revolution with timelike axis satisfying this case.
Let S be the surface of revolution in the Euclidean space [E.sup.3] with rotation axis z obtained from the above data.
From the well-known formulas for the Christoffel symbols of a surface of revolution, we deduce that Z is orthogonally Killing.
Note that if a is slightly less than 1, [S.sub.[alpha]] is very close to [S.sub.1] and does represent the area of the surface of revolution in question.
They are relevant here because the surface of revolution plots can be misleading.
13 displays the curve, the surface of revolution for m = 5 and [beta] = 0, and half the surface of revolution for b = 2[pi].
Typical applications include 2- and 3-D contours, ruled, swept and lofted surface, surface of revolution, projections of arbitrary surfaces and intersections of surfaces.

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