Revision #2493 → #3176 · back to history
addedHyperbolic geometryee0fd8bb25e6
addedHyperbolic parallel postulate369852b442ea
addedAt least two parallels through an external point3bdce92f9881
addedHyperbolic plane120348594e30
addedPseudospherical surface geometry50bb75fc17b4
addedSaddle surfaces have negative curvature5ef8ec52f239
addedHyperboloid model eventsd6a4d3baace1
addedEvents correspond to rapidities1941cbc59798
addedParallel postulate is only axiomatic difference012dda6d1d53
addedAbsolute geometry3c5314e4b7da
addedKinds of absolute geometry3aa5c2fc1345
addedAbsolute geometry theorems hold863e4a86b672
addedEuclid I.27-I.28 parallel linesbf2ebfa67079
addedEuclidean equivalences may fail hyperbolicallya1cea00b647d
addedAbsolute scaleab449a675c3a
addedSingle-line properties match Euclidean geometry89040750216e
addedTwo points define a lineb5ed10349cdc
addedLine segments extend infinitely3355c6cf9876
addedIntersecting-line properties match Euclidean geometry7cb81b6f44ab
addedDistinct lines intersect at most once3d5272313750
addedIntersecting lines form equal opposite angles13d8575fac4b
addedAdjacent angles are supplementary2981288820dc
addedMany lines avoid two intersecting lines6556b3d45451
addedLine properties are model-independentaa92a40cef0b
addedNon-intersecting lines differ from Euclidean casece1e5a68c73f
addedAt least two non-intersecting lines8354853d6515
addedInfinitely many hyperbolic parallels1ae868a521f0
addedTwo classes of non-intersecting linese690ecd847e6
addedLimiting parallels8e46d7963c7c
addedUltraparallel linesdcc5f7e4bed1
addedParallel lines as limiting parallels53a774cf7133
addedAngle of parallelism5b96a80a038e
addedUltraparallel theorem339035866b06
addedCircle circumference lower bound5e2613d1f267
addedHyperbolic Gaussian curvature is negative61ad6f7b7c66
addedCircle circumference formulaebb9da71fff7
addedDisk area formulac80175433859
addedCircumference-to-radius ratiof4916778ce25
addedSmall-circle Euclidean limit841304ccadf1
addedCircle geodesic curvature formula81175884256c
addedNo equidistant lineadf70a748bff
addedHypercyclebe86d273536a
addedHorocyclef504bd6c88a9
addedTwo horocycles through two pointsb9951d4ccaac
addedHorocycle centers for two points358983564f40
addedThree-point curve classification58175d031b2d
addedLine-segment lengtha0a3eb95ebf4
addedHypercycle arc length boundsc3a9e1382f89
addedHorocycle arc length equality and bounds47a7cafa23f6
addedHorocycle geodesic curvature1e0d16e1b67d
addedHypercycle geodesic curvature80c2c787565d
addedHyperbolic triangle angle sum1544be0e8111
addedTriangle defect389fc21bcad9
addedConvex hyperbolic polygon defect6e0466cdbbca
addedHyperbolic triangle area by defect75d73e291f18
addedConvex hyperbolic polygon area by defect2defbdca5178
addedHyperbolic triangle area bound90e202750a7b
addedIdeal triangle maximal area11cf33a89bc9
addedHyperbolic triangle incircledb6d6b7b4536
addedNo circumcircle for horocyclic or hypercyclic vertices979330d2d9d7
addedHyperbolic AAA congruenceadb86577840a
addedRegular apeirogon and pseudogone44c1f9ac535
addedEuclidean apeirogon construction degenerates6cb28bda07c6
addedHyperbolic apeirogon or pseudogon side lengthsff94a2c45498
addedApeirogon via limiting bisectors7594b6dd6dc7
addedApeirogon horocycle inscription23784be58789
addedPseudogon via diverging bisectorsadf6b3e1b696
addedPseudogon hypercycle inscription5f366391d030
addedRegular tessellation of hyperbolic planeec0a4663651f
addedInfinitely many Schwarz-triangle uniform tilings0af055ddb69e
addedHyperbolic triangle group8c4ed80ae0cd
addedInfinitely many non-Schwarz uniform tilings70b2eda01ae0
addedQuadrilateral fundamental domains007e065db20e
addedStandardized curvature scale1e4e69e4d817
addedTriangle area at curvature minus one95c458c74fa2
addedHorocyclic sector area5fb78badf9d3
addedUnit horocycle arcc47d0f21f050
addedConcentric horocycle arc ratio2803c6915bf4
addedCoordinate-system difficulties92ba813b4cb0
addedQuadrilateral angle sumb00e3844a64e
addedNo equidistant lines9d65c018711c
addedHyperbolic rectangle boundary6f83f09743af
addedParallel transport rotationb4cad5265324
addedHyperbolic coordinate systemse799ab13f39a
addedLobachevsky coordinates9f522ac1f44c
addedLobachevsky x-coordinate704627d8f781
addedLobachevsky y-coordinate0ee015064fd4
addedHorocycle coordinate system793dfa2b9b1b
addedModel-based coordinates5e5e3e784a4d
addedCartesian-like coordinatese783457be9e0
addedCartesian-like x-coordinate80c0c986a3cf
addedCartesian-like y-coordinatea0a70bdc3c25
addedDistance formula93bb29a0b919
addedMetric tensor field5ca1be60c17b
addedStraight-line forms59eba655455b
addedUltraparallel line forma551d49b0652
addedNegative-side asymptotic line formeaee63e8c44b
addedPositive-side asymptotic line formc8aa78f54355
addedPerpendicular-intersecting line form019b919faa46
addedAngled-intersecting line form43f2f89dee52
addedLine equations bounded domain40d6a8644b0f
addedParallel postulate independencea35c7c0035a2
addedEarly quadrilateral theorems835df37646d4
addedHyperbolic functionsfe4bfdfff534
addedLambert triangle area computationdb48f17c84f1
addedHyperbolic geometry self-consistency9ce31ba2ab1c
addedBeltrami relative consistency1ab6497c2e63
addedCayley-Klein metric construction592b4de1de01
addedProjective transformations as isometries00e42fe7cc63
addedKlein Cayley absolute theorem36e08f31e1f3
addedHyperbolic geometry consistencyffb465aeb444
addedPossible spatial geometries7dea54112629
addedLobachevsky curvature lower bound conclusiona5849032a0b1
addedSphere-world thought experiment92d580cf2afa
addedGeometrization possibilities52c509bd8096
addedRelativity unifies space and timeb601fb22e46d
addedMinkowski geometry987b55f497d5
addedRelativistic curvature geometries5c5e9d088146
addedRapidity as hyperbolic anglef61824ba8368
addedKinematic geometry89f3cedc7dd0
addedVelocity space is hyperbolicf9cda39384b0
addedVelocity-space distancee17e739e0278
addedPseudospheres in Euclidean spacef38db294bb11
addedHilbert immersion theorem064d52b7ff25
addedNon-metric Euclidean modelsce6923716526
addedThurston paper model0505aa21a1a3
addedCrochet hyperbolic plane6a7fcd01488b
addedHyperbolic soccerball612f6a92ca85
addedHyperbolic quilt192d8f4f5a77
addedPseudospheres8356352980fb
addedTractoid model7de1d26763d0
addedNo full Euclidean embeddingc61067cd64e4
addedFour common models461607006cba
addedModels define hyperbolic plane8c5e3bee1aae
addedModels extend to higher dimensions1055a7590309
addedBeltrami-Klein model names15a9728a0974
addedBeltrami-Klein disk modelcc93eb3f60f9
addedBeltrami-Klein lines3a5694e7acdc
addedBeltrami-Klein ball model9a0ff94d0440
addedBeltrami-Klein higher-dimensional linesd228a51b24dc
addedBeltrami-Klein straight lines121efa186e3f
addedBeltrami-Klein angle distortion1796c7165465
addedBeltrami-Klein circles not circlese8e1b743535a
addedBeltrami-Klein distance870791f5775b
addedPoincare disk modelc6ff47b0ee3d
addedPoincare disk lines39893f2f3646
addedPoincare disk conformaled2080428f1d
addedPoincare disk isometries5aad43ee5455
addedPoincare disk circlesb7a58c97d6f3
addedEuclidean and hyperbolic circle centers differ876cbac6e88c
addedPoincare disk horocyclesc54238de24a9
addedPoincare disk hypercycles789cfcd5495e
addedPoincare half-plane modeld5fed83fdd14
addedBoundary line excluded589809f07f74
addedUpper half-plane realization56b1e07cf903
addedHalf-plane hyperbolic lines1cc1334bc360
addedRay interval length1b1a41d59ac7
addedRay length homothety invariancebd13ddb90354
addedHalf-plane conformal52ac8f570e7f
addedHalf-plane isometriese6dc7d32d238
addedHalf-plane as disk limit17bdc198d916
addedHyperboloid model6cd9d02ccb00
addedHyperboloid model in relativity35a4fe96f935
addedMinkowski 3-space spacetime model58c7715d2af5
addedHyperboloid as observer events6ccc815356db
addedHyperbolic distance as rapidityc4c3275d0d48
addedHyperboloid model higher dimensionc18d18df1883
addedHemisphere model role3403b6dc8dba
addedHemisphere model domain6908986884dc
addedHemisphere hyperbolic linesace1b779c97a
addedHemisphere projections yield modelsab9cebf23a2c
addedAll models same structure37cab3ecb2fc
addedModels as coordinate charts20520a8ce0f0
addedCharacteristic curvature of hyperbolic planebde0b86dc023
addedGeodesic invariance47f40af33249
addedHyperbolic geometry via geodesicsd97abea78cdd
addedNon-isometric Euclidean embedding2450da082044
addedInfinitely many charts8eeaf6c95061
addedModel transformations0d77cac269d5
addedGans model11d724b0d9ae
addedGans model as projection84e5447f73a9
addedGans model uses entire plane74774882fb50
addedGans model lineseae1a681edd6
addedConformal square model60dab573613e
addedConformal square finite extent64a3b1311929
addedConformal square points inside squaref94ec705762d
addedConformal square conformal5e7604a92562
addedBand model043b89e8944d
addedBand model preserved distance linef5ec49814842
addedBand model metriceebd842d5647
addedIsometry reflection decomposition63ccdff990ca
addedHigher-dimensional reflection decomposition46853d5b8342
addedIsometry classification5644980c3d53
addedIdentity isometry381396471454
addedInversion through a pointf118b041c20c
addedRotation around a normal point64d6b734da7a
addedHorolationbc85dd79fd66
addedTranslation along a linec18e2fe5ceca
addedReflection through a line0f1e2f90e875
addedGlide reflection4d456231cd51
addedEscher conformal disk modelcc3f513bc5cb
addedCircle Limit III white lines are hypercyclesceaa164b841a
addedNegative curvature visible by angle sums0ad0d552e9dd
addedCircle Limit III vertex angles49401b02f19a
addedEuclidean comparison angle sum816034b2bca9
addedTriangle angle sum from Escher tiling1b86713b65b4
addedExponential growthdba34f32aa7d
addedFish count grows exponentially5fa8bd37555b
addedBall area grows exponentially1c0f9c512ab1
addedCrocheted hyperbolic planes38d8e6eb3518
addedHyperRogue hyperbolic tilings9046ac3e25b7
addedHyperbolic geometry in all dimensions0e8864c80219
addedHyperbolic space as symmetric space30b33136b42b
addedHyperbolic space quotient90241e9d2d38
addedOrthogonal group action on Minkowski spaceaad0ade3aae9
addedOrthogonal group transitive action908c11f2b223
addedTimelike lines meet antipodal hyperboloid pointsd1f36fbdf175
addedLine-space model of hyperbolic n-spacecefc783823dc
addedStabilizer of a line6266279870f3
addedLinear-algebraic hyperbolic concepts7f9038f8fdd1
addedGeodesic paths as intersectionse5d9977927dd
addedDihedral angles by inner products6ad64562652b
addedReflection groups by matricesf089d929f4b2
addedSmall-dimensional exceptional isomorphisms92cb41e5a97e
addedDimension two Lie group isomorphisms45f70067dece
addedSymmetry groups act fractionally linearlyb74ade4bb9f7
addedCayley transformation conjugates symmetry groups5fdac0d7ad5d
addedDimension three boundary action55dd250196c8
addedStudying hyperbolic 3-isometries by matrices31cc165df7aa
addedParabolic transformations8d472435af00
addedUnipotent triangular representativesceb1b1e5f05d