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The 35 pages that link to Hyperbolic geometry, each with the reason it gives.

• Euclidean geometryCompared with: It replaces Euclid’s unique-parallel rule with a geometry of negative curvature.

• Pythagorean theoremBroader topic: Its right triangles follow hyperbolic trigonometric laws instead of the Euclidean theorem.

• Euclidean spaceCompared with: Its geometry cannot be represented by the global Euclidean metric.

• Non-Euclidean geometryBroader topic: It replaces the unique-parallel rule with infinitely many parallels.

• Spherical geometryCompared with: It reverses spherical geometry’s angle-sum behavior: triangles have less than 180 degrees.

• GeometryBroader topic: It provides one precise alternative to Euclidean geometry’s unique parallel line.

• Plane (geometry)Compared with: It replaces the Euclidean plane’s parallel postulate with a different rule.

• Parallel linesCompared with: It replaces Euclid’s unique parallel with many nonintersecting choices.

• Parallel postulateCompared with: It replaces the postulate’s unique parallel with infinitely many nonintersecting lines.

• Triangle geometryCompared with: Its triangles do not obey the Euclidean angle-sum and center results unchanged.

• Möbius transformationNarrower topic: Möbius transformations furnish isometries of standard models of the hyperbolic plane.

• Triangle areaCompared with: Triangle areas follow curvature-dependent relations rather than the Euclidean formula alone.

• Spherical trigonometryCompared with: It provides a contrasting curved geometry, where triangle angle sums fall below 180 degrees.

• Elliptic geometryCompared with: It shares the rejection of Euclid's postulate but has negative curvature and many parallels.

• HyperbolaCompared with: Despite the shared name, this geometry is not simply the study of hyperbola curves.

• Hyperbolic functionsNarrower topic: Its geometric structures use hyperbolic functions, but the functions also arise outside this geometry.

• Nikolai LobachevskyBroader topic: This is the geometry Lobachevsky developed from denying Euclid’s parallel postulate.

• Foundations of GeometryCompared with: It satisfies alternatives to Euclid's parallel postulate while retaining many other geometric axioms.

• Geometric proofBroader topic: Its alternative parallel axiom produces proofs unlike their Euclidean counterparts.

• Hilbert's axiomsCompared with: Its alternative parallel behavior directly contrasts with Hilbert's Euclidean parallel axiom.

• János BolyaiBroader topic: This is the non-Euclidean geometry Bolyai independently developed.

• PseudosphereNarrower topic: The pseudosphere realizes a portion of this geometry as a surface.

• Spherical excessCompared with: Its triangles have an angular deficit where spherical triangles have an excess.

• Flat torusCompared with: It provides the negative-curvature counterpart to the flat torus's Euclidean local geometry.

• H. S. M. CoxeterRelated: Coxeter’s study of hyperbolic tessellations linked abstract geometry with Escher’s visual patterns.

• Maryam MirzakhaniRelated: Hyperbolic surfaces provide the geometric setting for many of her results.

• Hyperbolic triangleNarrower topic: It supplies the plane and distance rules in which hyperbolic triangles are defined.

• Hilbert's theoremRelated: Its constant-curvature plane is the intrinsic model the theorem forbids immersing completely.

• Playfair's axiomCompared with: It rejects Playfair's uniqueness condition while retaining other Euclidean-style assumptions.

• Hyperbolic angleRelated: Hyperbolic functions and hyperbolic parametrizations recur in models of this geometry.

• Markov numberRelated: Lengths of curves on punctured-torus structures satisfy Markov-type relations.

• Caroline SeriesRelated: It is a central setting for Series’s work on geometric structures and group actions.

• Daina TaimiņaRelated: The geometry Taimiņa’s crocheted models make visible through expanding curved surfaces.

• Hjelmslev's theoremNarrower topic: The theorem depends on hyperbolic geometry’s fixed negative curvature.

• Law of cotangentsCompared with: Its triangles violate the angle-sum condition used to establish the law.