In geometry, a pseudosphere is a surface in ${\displaystyle \mathbb {R} ^{3}}$. It is the most famous example of a pseudospherical surface. A pseudospherical surface is a surface piecewise smoothly immersed in ${\displaystyle \mathbb {R} ^{3}}$ with constant negative Gaussian curvature. A "pseudospherical surface of radius R" is a surface in ${\displaystyle \mathbb {R} ^{3}}$ having curvature −1/R² at each point. Its name comes from the analogy with the sphere of radius R, which is a surface of curvature 1/R². Examples include the tractroid, Dini's surfaces, breather surfaces, and the Kuen surface.

The term "pseudosphere" was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry.

By "the pseudosphere", people usually mean the tractroid. The tractroid is obtained by revolving a tractrix about its asymptote. As an example, the (half) pseudosphere (with radius 1) is the surface of revolution of the tractrix parametrized by

It is a singular space (the equator is a singularity), but away from the singularities, it has constant negative Gaussian curvature and therefore is locally isometric to a hyperbolic plane.

The name "pseudosphere" comes about because it has a two-dimensional surface of constant negative Gaussian curvature, just as a sphere has a surface with constant positive Gaussian curvature. Just as the sphere has at every point a positively curved geometry of a dome the whole pseudosphere has at every point the negatively curved geometry of a saddle.

As early as 1693 Christiaan Huygens found that the volume and the surface area of the pseudosphere are finite, despite the infinite extent of the shape along the axis of rotation. For a given edge radius R, the area is 4πR² just as it is for the sphere, while the volume is ⁠2/3⁠πR³ and therefore half that of a sphere of that radius.

The pseudosphere is an important geometric precursor to mathematical fabric arts and pedagogy.

A line congruence is a 2-parameter families of lines in ${\displaystyle \mathbb {R} ^{3}}$. It can be written as$${\displaystyle X(u,v,t)=x(u,v)+tp(u,v)}$$where each pick of ${\displaystyle u,v\in \mathbb {R} }$ picks a specific line in the family.