Pseudosphere

A surface of revolution with constant negative Gaussian curvature, formed by rotating a tractrix around its asymptote. It provides a local model of the hyperbolic plane, though it cannot represent the entire plane smoothly in ordinary three-dimensional space.

Tractrix: A curve whose tangent segment between a point and a fixed line has constant length. Rotating this curve around its asymptote generates the pseudosphere.
Hyperbolic geometry: A geometry with constant negative curvature, in which a point has multiple nonintersecting lines parallel to a given line. The pseudosphere realizes a portion of this geometry as a surface.
Eugenio Beltrami: An Italian mathematician who developed models demonstrating the consistency of non-Euclidean geometry. Beltrami used the pseudosphere to illustrate a concrete region of hyperbolic geometry.
Differential geometry: The study of smooth spaces using calculus, including their curvature, metrics, and geodesics. The pseudosphere is a standard example connecting surface shape to intrinsic geometry.
Gaussian curvature: A measure of intrinsic surface curvature equal to the product of the principal curvatures at a point. The pseudosphere has the same negative Gaussian curvature at every regular point.
Hyperbolic plane: The simply connected, complete two-dimensional space of constant negative curvature. The pseudosphere locally matches its geometry but omits regions needed for a complete model.
Nikolai Lobachevsky: A Russian mathematician who independently developed hyperbolic geometry in the nineteenth century. His geometry supplied the non-Euclidean framework that the pseudosphere could model locally.
Non-Euclidean geometry: Geometries that differ from Euclidean geometry, including spherical and hyperbolic geometry. The pseudosphere offers a spatial demonstration of one non-Euclidean geometry.
Surface of revolution: A surface formed by rotating a plane curve around an axis in its plane. The pseudosphere belongs to this class, with its shape fixed by the tractrix profile.
Geodesic: A curve that locally minimizes distance and generalizes a straight line to a curved space. Geodesics on the pseudosphere illustrate how straightest paths behave in negative curvature.

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Daina TaimiņaRelated: It is a classical surface model related to the geometry made tangible by crochet.