Surface of revolution


A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) one full revolution around an axis of rotation (normally not intersecting the generatrix, except at its endpoints).[1] The volume bounded by the surface created by this revolution is the solid of revolution.

A portion of the curve x = 2 + cos(z) rotated around the z-axis
A torus as a square revolved around an axis along the diagonal of the square.

Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. A circle that is rotated around any diameter generates a sphere of which it is then a great circle, and if the circle is rotated around an axis that does not intersect the interior of a circle, then it generates a torus which does not intersect itself (a ring torus).

Properties edit

The sections of the surface of revolution made by planes through the axis are called meridional sections. Any meridional section can be considered to be the generatrix in the plane determined by it and the axis.[2]

The sections of the surface of revolution made by planes that are perpendicular to the axis are circles.

Some special cases of hyperboloids (of either one or two sheets) and elliptic paraboloids are surfaces of revolution. These may be identified as those quadratic surfaces all of whose cross sections perpendicular to the axis are circular.

Area formula edit

If the curve is described by the parametric functions x(t), y(t), with t ranging over some interval [a,b], and the axis of revolution is the y-axis, then the surface area Ay is given by the integral


provided that x(t) is never negative between the endpoints a and b. This formula is the calculus equivalent of Pappus's centroid theorem.[3] The quantity


comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. The quantity x(t) is the path of (the centroid of) this small segment, as required by Pappus' theorem.

Likewise, when the axis of rotation is the x-axis and provided that y(t) is never negative, the area is given by[4]


If the continuous curve is described by the function y = f(x), axb, then the integral becomes


for revolution around the x-axis, and


for revolution around the y-axis (provided a ≥ 0). These come from the above formula.[5]

This can also be derived from multivariable integration. If a plane curve is given by   then its corresponding surface of revolution when revolved around the x-axis has Cartesian coordinates given by   with  . Then the surface area is given by the surface integral


Computing the partial derivatives yields



and computing the cross product yields


where the trigonometric identity   was used. With this cross product, we get



where the same trigonometric identity was used again. The derivation for a surface obtained by revolving around the y-axis is similar.

For example, the spherical surface with unit radius is generated by the curve y(t) = sin(t), x(t) = cos(t), when t ranges over [0,π]. Its area is therefore


For the case of the spherical curve with radius r, y(x) = r2x2 rotated about the x-axis


A minimal surface of revolution is the surface of revolution of the curve between two given points which minimizes surface area.[6] A basic problem in the calculus of variations is finding the curve between two points that produces this minimal surface of revolution.[6]

There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid.[7]

Coordinate expressions edit

A surface of revolution given by rotating a curve described by   around the x-axis may be most simply described by  . This yields the parametrization in terms of   and   as  . If instead we revolve the curve around the y-axis, then the curve is described by  , yielding the expression   in terms of the parameters   and  .

If x and y are defined in terms of a parameter  , then we obtain a parametrization in terms of   and  . If   and   are functions of  , then the surface of revolution obtained by revolving the curve around the x-axis is described by  , and the surface of revolution obtained by revolving the curve around the y-axis is described by  .

Geodesics edit

Meridians are always geodesics on a surface of revolution. Other geodesics are governed by Clairaut's relation.[8]

Toroids edit

A toroid generated from a square

A surface of revolution with a hole in, where the axis of revolution does not intersect the surface, is called a toroid.[9] For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow square-section ring is produced. If the revolved figure is a circle, then the object is called a torus.

Applications edit

The use of surfaces of revolution is essential in many fields in physics and engineering. When certain objects are designed digitally, revolutions like these can be used to determine surface area without the use of measuring the length and radius of the object being designed.

See also edit

References edit

  1. ^ Middlemiss; Marks; Smart. "15-4. Surfaces of Revolution". Analytic Geometry (3rd ed.). p. 378. LCCN 68015472.
  2. ^ Wilson, W.A.; Tracey, J.I. (1925), Analytic Geometry (Revised ed.), D.C. Heath and Co., p. 227
  3. ^ Thomas, George B. "6.7: Area of a Surface of Revolution; 6.11: The Theorems of Pappus". Calculus (3rd ed.). pp. 206–209, 217–219. LCCN 69016407.
  4. ^ Singh, R.R. (1993). Engineering Mathematics (6 ed.). Tata McGraw-Hill. p. 6.90. ISBN 0-07-014615-2.
  5. ^ Swokowski, Earl W. (1983), Calculus with analytic geometry (Alternate ed.), Prindle, Weber & Schmidt, p. 617, ISBN 0-87150-341-7
  6. ^ a b Weisstein, Eric W. "Minimal Surface of Revolution". MathWorld.
  7. ^ Weisstein, Eric W. "Catenoid". MathWorld.
  8. ^ Pressley, Andrew. “Chapter 9 - Geodesics.” Elementary Differential Geometry, 2nd ed., Springer, London, 2012, pp. 227–230.
  9. ^ Weisstein, Eric W. "Toroid". MathWorld.

External links edit

  • Weisstein, Eric W. "Surface of Revolution". MathWorld.
  • "Surface de révolution". Encyclopédie des Formes Mathématiques Remarquables (in French).