Pseudosphere

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.