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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Möbius transformations canonical | 5 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991667 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Möbius transformations Context triple: [Kleinian group, actsBy, Möbius transformations]
-
A.
Möbius geometry
Möbius geometry is a branch of geometry that studies properties of figures invariant under Möbius (conformal) transformations of the extended complex plane or higher-dimensional spheres.
-
B.
Riemann sphere
The Riemann sphere is the complex plane plus a point at infinity, forming a one-dimensional complex manifold topologically equivalent to a sphere and used to study meromorphic functions and complex analysis.
-
C.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
-
D.
Poincaré upper half-plane model
The Poincaré upper half-plane model is a standard representation of the hyperbolic plane using the complex numbers with positive imaginary part, equipped with a specific metric that makes geodesics appear as semicircles and vertical lines.
-
E.
Transformations
"Transformations" is an influential art-critical work by British critic Roger Fry that explores the nature and evolution of modern art and aesthetic experience.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Möbius transformations Target entity description: Möbius transformations are conformal automorphisms of the extended complex plane represented by fractional linear functions that map circles and lines to circles and lines.
-
A.
Möbius geometry
Möbius geometry is a branch of geometry that studies properties of figures invariant under Möbius (conformal) transformations of the extended complex plane or higher-dimensional spheres.
-
B.
Riemann sphere
The Riemann sphere is the complex plane plus a point at infinity, forming a one-dimensional complex manifold topologically equivalent to a sphere and used to study meromorphic functions and complex analysis.
-
C.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
-
D.
Poincaré upper half-plane model
The Poincaré upper half-plane model is a standard representation of the hyperbolic plane using the complex numbers with positive imaginary part, equipped with a specific metric that makes geodesics appear as semicircles and vertical lines.
-
E.
Transformations
"Transformations" is an influential art-critical work by British critic Roger Fry that explores the nature and evolution of modern art and aesthetic experience.
- F. None of above. chosen
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)
ⓘ
linked to:
SL(2,C)
|
| characterizedBy | images of three distinct points ⓘ |
| definedOn |
Riemann sphere
ⓘ
extended complex plane ⓘ |
| determinedBy | values at three distinct points ⓘ |
| formsGroupUnder | composition ⓘ |
| groupIsIsomorphicTo |
PGL(2,ℂ)
ⓘ
PSL(2,ℂ) ⓘ
linked to:
PSL(2,\mathbb{C})
|
| 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 ⓘ |
| 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Möbius transformations Description of subject: Möbius transformations are conformal automorphisms of the extended complex plane represented by fractional linear functions that map circles and lines to circles and lines.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.