KnowraHyperbolic geometryLinked fromLinked fromThe 35 pages that link to Hyperbolic geometry, each with the reason it gives.All 35Broader topic 6Related 7Narrower topic 5Compared with 17Euclidean geometryCompared with: It replaces Euclid’s unique-parallel rule with a geometry of negative curvature.Euclidean spaceCompared with: Its geometry cannot be represented by the global Euclidean metric.Spherical geometryCompared with: It reverses spherical geometry’s angle-sum behavior: triangles have less than 180 degrees.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.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.Foundations of GeometryCompared with: It satisfies alternatives to Euclid's parallel postulate while retaining many other geometric axioms.Hilbert's axiomsCompared with: Its alternative parallel behavior directly contrasts with Hilbert's Euclidean parallel axiom.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.Playfair's axiomCompared with: It rejects Playfair's uniqueness condition while retaining other Euclidean-style assumptions.Law of cotangentsCompared with: Its triangles violate the angle-sum condition used to establish the law.
KnowraHyperbolic geometryLinked fromLinked fromThe 35 pages that link to Hyperbolic geometry, each with the reason it gives.All 35Broader topic 6Related 7Narrower topic 5Compared with 17Euclidean geometryCompared with: It replaces Euclid’s unique-parallel rule with a geometry of negative curvature.Euclidean spaceCompared with: Its geometry cannot be represented by the global Euclidean metric.Spherical geometryCompared with: It reverses spherical geometry’s angle-sum behavior: triangles have less than 180 degrees.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.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.Foundations of GeometryCompared with: It satisfies alternatives to Euclid's parallel postulate while retaining many other geometric axioms.Hilbert's axiomsCompared with: Its alternative parallel behavior directly contrasts with Hilbert's Euclidean parallel axiom.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.Playfair's axiomCompared with: It rejects Playfair's uniqueness condition while retaining other Euclidean-style assumptions.Law of cotangentsCompared with: Its triangles violate the angle-sum condition used to establish the law.