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Surfaces of Revolution with Constant Gaussian Curvature

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A surface of revolution arises by rotating a curve in the - plane around the axis. This Demonstration lets you explore the surfaces of revolution with constant nonzero Gaussian curvature . Without loss of generality, we restrict ourselves to surfaces with and (other curvatures are obtained by an appropriate scaling).

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Surfaces with form a one-parameter family; use the slider to change the parameter value (the value 1 corresponds to the sphere). The Gaussian curvature is the product of the principal curvatures, so as one increases, the other decreases in such a way that their product remains constant.

There are three types of surfaces with ; the first is symmetric with respect to a horizontal plane; the second one is the pseudosphere, whose profile curve (called tractrix) has the axis as an asymptote; and the third type has a profile curve that reaches the axis in a finite time. The surfaces in the first and third classes are again controlled by a parameter.

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Contributed by: Antonin Slavik (May 2011)
(Charles University, Prague)
Open content licensed under CC BY-NC-SA

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Details

Assume that the profile curve is parametrized by arc length, that is, . Gaussian curvature of the corresponding surface of revolution is then given by . Since is a constant, we obtain a second-order differential equation . This equation is easy to solve analytically; the corresponding function  is then calculated from the first equation by numerical integration.

See, for example, Section 5.7 in [1] or Chapter 3 of [2].

References

[1] B. O’Neill, Elementary Differential Geometry, 2nd ed., London: Elsevier Academic Press, 2006.

[2] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 3, 3rd ed., Houston: Publish or Perish, 1999.

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