Surface of Revolution

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A surface of revolution is a surface generated by rotating a two-dimensional curve about an axis. The resulting surface therefore always has azimuthal symmetry. Examples of surfaces of revolution include the apple surface, cone (excluding the base), conical frustum (excluding the ends), cylinder (excluding the ends), Darwin-de Sitter spheroid, Gabriel's horn, hyperboloid, lemon surface, oblate spheroid, paraboloid, prolate spheroid, pseudosphere, sphere, spheroid, and torus (and its generalization, the toroid).

The area element of the surface of revolution obtained by rotating the curve y=f(x)>0 from x=a to x=b about the x-axis is

so the surface area is

(Apostol 1969, p. 286; Kaplan 1992, p. 251; Anton 1999, p. 380). If the curve is instead specified parametrically by (x(t),y(t)), the surface area obtained by rotating the curve about the x-axis for t in [a,b] if x(t)>0 in this interval is given by

Similarly, the area of the surface of revolution obtained by rotating the curve x=g(y)>0 from y=c to y=d about the y-axis is given by

(Anton 1999, p. 380). If the curve is instead specified parametrically by (x(t),y(t)), the surface area obtained by rotating the curve about the y-axis for t in [c,d] if y(t)>0 in this interval is given by

The following table gives the lateral surface areas S for some common surfaces of revolution where r denotes the radius (of a cone, cylinder, sphere, or zone), R_1 and R_2 the inner and outer radii of a frustum, h the height, e the ellipticity of a spheroid, and a and c the equatorial and polar radii (for a spheroid) or the radius of a circular cross-section and rotational radius (for a torus).

The standard parameterization of a surface of revolution is given by

For a curve so parameterized, the first fundamental form has

Wherever phi and phi^('2)+psi^('2) are nonzero, then the surface is regular and the second fundamental form has

Furthermore, the unit normal vector is

and the principal curvatures are

The Gaussian and mean curvatures are

(Gray 1997).

Pappus's centroid theorem gives the volume of a solid of rotation as the cross-sectional area times the distance traveled by the centroid as it is rotated.

See also

Apple Surface, Catenoid, Cone, Conical Frustum, Cylinder, Darwin-de Sitter Spheroid, Eight Surface, Gabriel's Horn, Hyperboloid, Lemon Surface, Meridian, Minimal Surface of Revolution, Oblate Spheroid, Pappus's Centroid Theorem, Paraboloid, Peninsula Surface, Prolate Spheroid, Pseudosphere, Sinclair's Soap Film Problem, Solid of Revolution, Sphere, Spheroid, Surface of Revolution Parallel, Toroid, Torus, Unduloid Explore this topic in the MathWorld classroom

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References

Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, 1999.Apostol, T. M. Calculus, 2nd ed., Vol. 2: Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability. Waltham, MA: Blaisdell, 1969.Gray, A. "Surfaces of Revolution." Ch. 20 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 457-480, 1997.Hilbert, D. and Cohn-Vossen, S. "The Cylinder, the Cone, the Conic Sections, and Their Surfaces of Revolution." §2 in Geometry and the Imagination. New York: Chelsea, pp. 7-11, 1999.Kaplan, W. Advanced Calculus, 3rd ed. Reading, MA: Addison-Wesley, 1992.Kreyszig, E. Differential Geometry. New York: Dover, p. 131, 1991.

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Surface of Revolution

Cite this as:

Weisstein, Eric W. "Surface of Revolution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SurfaceofRevolution.html

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