In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) … See more

Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting ge… See more

Though hyperbolic geometry applies for any surface with a constant negative Gaussian curvature, it is usual to assume a scale in which the curvature K is −1.
This results in some formulas becoming simpler. So… See more

Since the publication of Euclid's Elements circa 300BC, many geometers tried to prove the parallel postulate. Some tried to prove it by assuming its negation and trying to derive a contradiction. Foremost among these were … See more

There exist various pseudospheres in Euclidean space that have a finite area of constant negative Gaussian curvature.
By Hilbert's theorem, one cannot isometrically immerse a complete hyperbolic plane (a co… See more

Various pseudospheres – surfaces with constant negative Gaussian curvature – can be embedded in 3-D space under the standard Euclidean metric, and so can be made into tangible models. Of these, the tractoid (or … See more