Möbius transformations
E898486
bijection of the Riemann sphere
complex function
conformal map
fractional linear transformation
rational function
Möbius transformations are conformal automorphisms of the extended complex plane represented by fractional linear functions that map circles and lines to circles and lines.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Möbius transformation | 0 |
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
bijection of the Riemann sphere
ⓘ
complex function ⓘ conformal map ⓘ fractional linear transformation ⓘ rational function ⓘ |
| actsOn |
boundary of hyperbolic 3-space
ⓘ
unit disk ⓘ upper half-plane ⓘ |
| associatedMatrixGroup | SL(2,ℂ) GENERATED ⓘ |
| characterizedBy | images of three distinct points ⓘ |
| definedOn |
Riemann sphere
NERFINISHED
ⓘ
extended complex plane ⓘ |
| determinedBy | values at three distinct points ⓘ |
| formsGroupUnder | composition ⓘ |
| groupIsIsomorphicTo |
PGL(2,ℂ)
NERFINISHED
ⓘ
PSL(2,ℂ) NERFINISHED ⓘ |
| hasGeneralForm | f(z) = (az + b) / (cz + d) ⓘ |
| hasInverse | another Möbius transformation ⓘ |
| hasParameter |
a ∈ ℂ
ⓘ
b ∈ ℂ ⓘ c ∈ ℂ ⓘ d ∈ ℂ ⓘ |
| hasPoleAt | z = −d/c when c ≠ 0 ⓘ |
| isBiholomorphism | of the Riemann sphere ⓘ |
| isBijective | true ⓘ |
| isConformal | true ⓘ |
| isGeneratedBy |
dilations and rotations
ⓘ
inversions ⓘ translations ⓘ |
| isHolomorphicExcept | at most one point ⓘ |
| isRationalOfDegree | 1 ⓘ |
| maps |
circles to circles or lines
ⓘ
extended complex plane to itself ⓘ lines to circles or lines ⓘ |
| matrixEquivalenceRelation | scalar multiples represent same transformation ⓘ |
| namedAfter | August Ferdinand Möbius NERFINISHED ⓘ |
| preserves |
angles
ⓘ
cross-ratio ⓘ orientation ⓘ |
| representedBy | 2×2 complex matrix [[a,b],[c,d]] ⓘ |
| satisfiesCondition | ad − bc ≠ 0 ⓘ |
| sends | −d/c to ∞ when c ≠ 0 ⓘ |
| sendsInfinityTo |
a/c when c ≠ 0
ⓘ
∞ when c = 0 ⓘ |
| specialCase |
z ↦ 1/z (inversion)
ⓘ
z ↦ az (dilation-rotation) ⓘ z ↦ z + b (translation) ⓘ |
| usedIn |
complex analysis
ⓘ
geometric function theory ⓘ hyperbolic geometry ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.