# Surfaces of Revolution with Constant Gaussian Curvature

Surfaces of Revolution with Constant Gaussian Curvature

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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K=1

K=-1

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.

K=1

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.

K=-1

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