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). 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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