A surface of revolution is a Surface generated by rotating a 2-D Curve about an axis. The resulting surface therefore always has azimuthal symmetry. Examples of surfaces of revolution include the Apple, Cone (excluding the base), Conical Frustum (excluding the ends), Cylinder (excluding the ends), Darwin-de Sitter Spheroid, Gabriel's Horn, Hyperboloid, Lemon, Oblate Spheroid, Paraboloid, Prolate Spheroid, Pseudosphere, Sphere, Spheroid, and Torus (and its generalization, the Toroid).

The standard parameterization of a surface of revolution is given by

(1) | |||

(2) | |||

(3) |

For a curve so parameterized, the first Fundamental Form has

(4) | |||

(5) | |||

(6) |

Wherever and are nonzero, then the surface is regular and the second Fundamental Form has

(7) | |||

(8) | |||

(9) |

Furthermore, the unit Normal Vector is

(10) |

(11) | |||

(12) |

The Gaussian and Mean Curvatures are

(13) | |||

(14) |

(Gray 1993).

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.

Calculus of Variations can be used to find the curve from a point to a point which, when
revolved around the *x*-Axis, yields a surface of smallest Surface Area (i.e., the Minimal Surface).
This is equivalent to finding the Minimal Surface passing through two circular wire frames. The Area element is

(15) |

(16) |

(17) |

(18) |

(19) |

(20) |

(21) |

(22) |

(23) |

(24) |

(25) |

(26) |

(27) | |||

(28) |

which cannot be solved analytically.

The general case is somewhat more complicated than this solution suggests. To see this, consider the Minimal Surface
between two rings of equal Radius . Without loss of generality, take the origin at the midpoint of the two rings.
Then the two endpoints are located at and , and

(29) |

(30) |

(31) |

(32) |

(33) |

To find the maximum value of at which Catenary solutions can be obtained, let . Then
(31) gives

(34) |

(35) |

(36) |

(37) |

(38) |

(39) |

(40) |

The Surface Area of the minimal Catenoid surface is given by

(41) |

(42) | |||

(43) |

(44) |

Some caution is needed in solving (33) for . If we take and then (33) becomes

(45) |

The Surface Area of the Catenoid solution equals that of the Goldschmidt Solution when (44) equals
the Area of two disks,

(46) |

(47) |

(48) |

(49) |

(50) |

(51) |

(52) |

(53) |

(54) |

For , the Catenary solution has larger Area than the two disks, so it exists only as a Relative Minimum.

There also exist solutions with a disk (of radius ) between the rings supported by two Catenoids of
revolution. The Area is larger than that for a simple Catenoid, but it is a Relative Minimum. The
equation of the Positive half of this curve is

(55) |

(56) |

(57) |

(58) |

Now let , so

(59) |

The Area of the central Disk is

(60) |

(61) |

(62) |

and

(63) |

(64) |

so

(65) |

(66) |

(67) |

(68) |

If we are interested instead in finding the curve from a point to a point which, when revolved around
the *y*-Axis (as opposed to the *x*-Axis), yields a surface of smallest
Surface Area , we proceed as above. Note that the solution is physically equivalent to that for rotation about the
*x*-Axis, but takes on a different mathematical form. The Area element is

(69) |

(70) |

(71) |

(72) | |||

(73) |

so the Euler-Lagrange Differential Equation becomes

(74) |

(75) |

(76) |

(77) |

(78) |

(79) |

(80) |

(81) |

Isenberg (1992, p. 80) discusses finding the Minimal Surface passing through two rings with axes offset from each other.

**References**

Arfken, G. *Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 931-937, 1985.

Goldstein, H. *Classical Mechanics, 2nd ed.* Reading, MA: Addison-Wesley, p. 42, 1980.

Gray, A. ``Surfaces of Revolution.'' Ch. 18 in *Modern Differential Geometry of Curves and Surfaces.*
Boca Raton, FL: CRC Press, pp. 357-375, 1993.

Isenberg, C. *The Science of Soap Films and Soap Bubbles.* New York: Dover, pp. 79-80 and Appendix III, 1992.

© 1996-9

1999-05-26