Linear Fractional Transformation

GPTKB entity

Predicate	Object
gptkbp:instanceOf	gptkb:mathematical_concept
gptkbp:alsoKnownAs	gptkb:Möbius_transformation
gptkbp:application	gptkb:geometry gptkb:signal_processing gptkb:Control_theory Conformal mapping
gptkbp:condition	ad - bc ≠ 0
gptkbp:definedIn	f(z) = (az + b) / (cz + d), where ad - bc ≠ 0
gptkbp:designer	Also a linear fractional transformation
gptkbp:domain	Extended complex plane
gptkbp:example	f(z) = (z-1)/(z+1) f(z) = (z-2)/(3z+4) f(z) = 1/z f(z) = az + b
gptkbp:field	gptkb:Mathematics Complex analysis
gptkbp:fixedPoints	May have 0, 1, or 2 fixed points
gptkbp:groupStructure	Forms a group under composition
gptkbp:hasSpecialCase	translator Inversion Rotation Dilation Affine transformation
https://www.w3.org/2000/01/rdf-schema#label	Linear Fractional Transformation
gptkbp:invertible	Yes
gptkbp:mapType	Circles to circles Lines to circles
gptkbp:matrixRepresentation	2x2 complex matrix
gptkbp:namedAfter	gptkb:August_Ferdinand_Möbius
gptkbp:parameter	a b c d
gptkbp:preserves	gptkb:Cross-ratio Circles and lines
gptkbp:range	Extended complex plane
gptkbp:relatedGroup	gptkb:Projective_linear_group_PGL(2,_C)
gptkbp:relatedTo	gptkb:Automorphism_of_the_Riemann_sphere gptkb:Bilinear_transformation gptkb:Projective_geometry
gptkbp:usedIn	gptkb:Riemann_sphere gptkb:Complex_projective_line Fractional linear system
gptkbp:bfsParent	gptkb:Conformal_Mapping
gptkbp:bfsLayer	7