Revision #605 → #1409 · back to history
addedMöbius transformation52def74d1ecf
addedGeometric construction via stereographic projection8b5459e28be7
addedAngle and generalized-circle preservation368211685b77
addedMöbius group as PGL(2,C)712a24c62299
addedDomain on extended complex plane726095c06423
addedMöbius transformations are automorphisms of the Riemann sphere10af79553039
addedIsomorphism with isometries of hyperbolic 3-spacee5dcc00dd1ff
addedLiouville's theorem (higher dimensions)174e5773c086
addedGeneral form of a Möbius transformationa594be4c9bb5
addedExtension when c ≠ 0c2036d104512
addedExtension when c = 00cd7416a337a
addedBijective holomorphic map of the Riemann sphere187e482f4af5
addedMöbius group as a complex Lie group3f2ab386fb16
addedDegenerate case ad = bc749b317ccbfa
addedKernel of the Schwarzian derivative4520162824dd
addedExistence of two fixed pointsb0c96aa5bab1
addedFixed-point formula via quadratic90c6ee8003f3
addedLinear case when c = 021679a930f59
addedSimple transformation: translations, rotations, dilationsc23d1cd183dd
addedPure translation when c = 0 and a = d1045915e877a
addedTopological proof via Euler characteristica4b93ef5b57c
addedPGL(2,K) is sharply 3-transitive2ec1eb92bf3f
addedLefschetz–Hopf fixed-point theorem22fd82b9b516
addedContrast with PGL(2,R)b9e7fcf128b1
addedNon-parabolic conjugate to dilation/rotation4b66e8228f76
addedCharacteristic constant (multiplier)9d76e4ed974b
addedRepulsive and attractive fixed points (loxodromic)6c11138d2135
addedParabolic normal form as translation12852f99645a
addedTranslation length6c11b07b0a43
addedPole of a Möbius transformationa01665685e62
addedInverse poleb712175ffb22
addedCharacteristic parallelogram99f6cbc921ee
addedTransform specified by two fixed points and polee268e9a68510
addedDecomposition as sequence of simple transformationsed3dc938078b
addedSimple transformations: translation, rotation, homothety, inversionec492fad7c53
addedExplicit composition decompositiona54deffb267f
addedEquivalence to sequence of simpler transformations77dfe9cd6585
addedFormula for inverse via composition73f96302f203
addedPreservation of anglesae36c5b4f1ea
addedGeneralized circles mapped to generalized circles329967d9cd0d
addedCross-ratios are invariant4481c6c5a2f1
addedCross-ratio at infinity by limitf0f964e9e8e0
addedCross-ratio real iff concyclic/collinearb082a2d34cca
addedConjugate points w.r.t. a generalized circlebc0ad5d975bc
addedConjugate w.r.t. a line (symmetric)7b6c535a3158
addedConjugate w.r.t. a circle (inversion)3c880f33d606
addedMöbius transformations preserve conjugation29c0023c7af2
addedNatural action of PGL(2,C) equals Möbius action8ba22f7a66c5
addedIdentification of CP^1 with the Riemann spherea34c72d70416
addedAction of PGL(2,C) on the projective lined92e154c15af
addedGroup isomorphism Möbius ≅ PGL(2,C)fba68b80c43a
addedPGL(2,K) as fractional linear transformations87d9b48b59ef
addedSL(2,C) → PGL(2,C) surjection with kernel ±I5d26ea81d558
addedMöbius group is a semisimple non-compact Lie group with π₁ = Z₂bdfeaebf3f90
addedSharp 3-transitivity: unique Möbius transformation from three points3a0e6bb8d481
addedMapping three points to 0, 1, ∞0db7ec6cfb14
addedAnharmonic group9183111991a1
addedExplicit determinant formula via hyperbola785aa696e716
addedPSL(2,R) as upper half-plane stabilizer8e4382b7fd7f
addedOpen-disk-preserving subgroup1a2c6ac1a837
addedIsomorphism between half-plane and disk subgroups6a1513375da7
addedMaximal compact subgroup is PSU(2) ≅ SO(3)d2cf71241065
addedIcosahedral groups and the quintic0b628f6bb1b1
addedModular group PSL(2,Z) and Fuchsian groups7015ebe65815
addedClassification into four types7ea35ee2231c
addedTrace invariant under conjugation; conjugacy criterion6e2e11cc1f82
addedParabolic transformationb7c44f977099
addedParabolic iff exactly one fixed pointb42329f21a85
addedParabolic subgroup as unipotent radicalfdd3e72a9e2e
addedCharacteristic constant for non-parabolicbf64b4eb974f
addedElliptic transformation8b1f51d74b8e
addedElliptic iff |λ| = 1 and λ ≠ ±15fc54e94f44a
addedCircular transform and three transpositions fixing {0,1,∞}39938a9f98a4
addedHyperbolic transformation4ddc58efdfad
addedHyperbolic iff λ is real and λ ≠ ±1709113db0201
addedLoxodromic transformation0bf2636b1452
addedClassification over the realsdf8002981fd7
addedLogarithmic form of the characteristic constant951d19ad8918
addedElliptic: ρ = 0, indifferent fixed points7c2b7079b66c
addedElliptic one-parameter subgroup along nested circles5f51ce5a454d
addedPhysical interpretation: rotating observer09d242646e1f
addedHyperbolic: α = 0, motion between fixed points9fa0d9bc1c1b
addedHyperbolic one-parameter subgroup along circular arcsc3488695f486
addedPhysical interpretation: accelerating observer70a19a28e8a9
addedLoxodromic: both nonzero, S-shaped paths033e53d4ca47
addedLoxodromic one-parameter subgroup as spiral16709ee99f6c
addedPhysical interpretation: rotating and translating observer57e963e05885
addedIteration formula via characteristic constant8b6e5d03831d
addedMöbius transformation in higher dimensions0acc606f6462
addedLiouville's theorem in conformal geometrye50940bb583c
addedOrientation-preserving form connected component3d4b62c9f206
addedMöbius geometry on the n-sphere988e93087be1
addedIsomorphism with restricted Lorentz group57ff4d90f936
addedMinkowski space with quadratic form11ee6e07fec5
addedTimelike, spacelike, null cone, celestial sphereaf0014522175
addedSO+(1,3) ≅ PSL(2,C) via hermitian matrices71a8494311c9
addedMöb(n) and isomorphism with SO+(1,n+1)08e82ab45881
addedMöbius group equals isometries of hyperbolic 3-spacefcb2d0974a0d