Statements (60)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
watch
|
| gptkbp:defines |
a function of the form f(z) = (az + b) / (cz + d)
|
| gptkbp:hasInventor |
f^(-1)(w)_=_(dw_-_b)_/_(-cw_+_a)
|
| gptkbp:hasSpecialty |
maps circles to circles or lines
preserves angles |
| https://www.w3.org/2000/01/rdf-schema#label |
Mœbius transformation
|
| gptkbp:isA |
bijective function
|
| gptkbp:isAvenueFor |
computer graphics
the field of optics the study of fractals the field of mathematical physics the field of signal processing the field of computational geometry |
| gptkbp:isCharacterizedBy |
determinant ad - bc ≠ 0
its ability to map lines to lines its action on the extended complex plane its effect on the topology of the plane its preservation of cross-ratios its_invariance_under_Möbius_transformations. |
| gptkbp:isInvolvedIn |
the mapping of the unit disk to itself
|
| gptkbp:isLocatedIn |
terms of cross-ratios
|
| gptkbp:isNamedAfter |
August Ferdinand Mœbius
|
| gptkbp:isRelatedTo |
gptkb:the_Poincaré_disk_model
the concept of duality the concept of infinity conformal mappings Riemann surfaces the concept of symmetry the concept of continuity its inverse transformation the concept of homogeneity the concept of projective transformations the concept of stereographic projection the concept of transformation groups the study of hyperbolic geometry |
| gptkbp:isRepresentedIn |
a 2x2 matrix
|
| gptkbp:isUsedBy |
analyze dynamical systems
transform the extended complex plane |
| gptkbp:isUsedFor |
a composition of translations, dilations, rotations, and inversions
|
| gptkbp:isUsedIn |
projective geometry
complex analysis the field of mathematics the study of analytic functions the study of algebraic topology the study of modular forms the study of differential equations the study of complex functions the study of mathematical modeling the study of algebraic curves the study of complex manifolds the study of rational functions the study of numerical analysis the study of complex dynamics the analysis of conformal maps the analysis of geometric transformations the design of certain types of filters the study of fractal geometry the study of geometric function theory the theory of functions of a complex variable |
| gptkbp:location |
a, b, c, d are complex numbers
|