Surfaces of Revolution with Constant Gaussian Curvature

surface type

positive curvature

a

A surface of revolution arises by rotating a curve in the

x

-

z

plane around the

z

axis. This Demonstration lets you explore the surfaces of revolution with constant nonzero Gaussian curvature

K

. Without loss of generality, we restrict ourselves to surfaces with

K=1

and

K=-1

(other curvatures are obtained by an appropriate scaling).

Surfaces with

K=1

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

K=-1

; the first is symmetric with respect to a horizontal plane; the second one is the pseudosphere, whose profile curve (called tractrix) has the

z

axis as an asymptote; and the third type has a profile curve that reaches the

z

axis in a finite time. The surfaces in the first and third classes are again controlled by a parameter.