In mathematics, a linear fractional transformation is, roughly speaking, an invertible transformation of the form z ↦ a z + b c z + d . {\displaystyle...

transformation is a linear fractional transformation that is not an affine transformation. Thus the generation of a special conformal transformation involves...

cases in which w = f(z) is a constant. The linear fractional transformation, also known as a Möbius transformation, has many fascinating properties. Four...

homographies, linear fractional transformations, bilinear transformations, and spin transformations (in relativity theory). Möbius transformations are defined...

linear-fractional programming (LFP) is a generalization of linear programming (LP). Whereas the objective function in a linear program is a linear function...

(mathematics). Affine transformation, in geometry Linear transformation between modules in linear algebra. Also called a linear map. Transformation matrix which...

in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It...

projective transformations, homographies, or linear fractional transformations. They form the projective linear group PGL(2, R). Each element of PGL(2, R)...

{\displaystyle z\mapsto {\frac {za+b}{zc+d}}\ ,} but otherwise the linear fractional transformation is seen as an equivalence: U [ z a + b ,   z c + d ] ∼ U [...

modular group acts on the upper-half of the complex plane by linear fractional transformations. The name "modular group" comes from the relation to moduli...

PSL(2,R) acts on H {\displaystyle H} by linear fractional transformations (also known as Möbius transformations): ( a b c d ) ⋅ z = a z + b c z + d . {\displaystyle...

numbers, and dual numbers. The conformal maps are described by linear fractional transformations in each case. In Riemannian geometry, two Riemannian metrics...

fractional Fourier, Laplace, Gauss–Weierstrass, Bargmann and the Fresnel transforms as particular cases. The name "linear canonical transformation" is...

the proof. If f ( x ) {\displaystyle f(x)} is given by the linear fractional transformation f ( x ) = ( cos ⁡ α ) x − sin ⁡ α ( sin ⁡ α ) x + cos ⁡ α ...

necessarily centerless). As for Möbius transformations, the group PGL(2, K) can be interpreted as fractional linear transformations with coefficients in K. Points...

taken to be complex numbers, any Möbius transformation is represented by a linear fractional transformation of complex numbers, and the hyperbolic motions...

(also called linear fractional transformations), with the understanding that when the denominator of the linear fractional transformation is 0, the image...

one, then there are exactly zero or one positive roots. A linear fractional transformation of the variable makes it possible to use the rule of signs...

variable linear fractional transformations just as circular angle is preserved by the Möbius transformations of the ordinary complex plane. Transformations preserving...

{\text{ for }}\{z:|z|<R*\}} Results for compositions of linear fractional (Möbius) transformations include the following, as examples: Theorem LFT1—On the...

the name anharmonic ratio. The cross-ratio is preserved by linear fractional transformations. It is essentially the only projective invariant of a quadruple...

correspondence with oriented lines on the plane. A Laguerre transformation is a linear fractional transformation z ↦ a z + b c z + d {\displaystyle z\mapsto {\frac...

Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number...

that is, a discrete subgroup Γ of PSL(2, R) – acts on H via linear fractional transformation. Each element of PSL(2, R) defines an isometry of H, so Γ is...

is linear in xi combined with a link function, this is not strictly necessary. The first approach uses a log-odds transformation of y as a linear function...

different methodologies to design a LPV controller namely, Linear fractional transformations which relies on the small gain theorem for bounds on performance...

be substituted for u−1. Homographies on P1(A) are called linear-fractional transformations since U [ z , 1 ] ( a c b d ) = U [ z a + b , z c + d ] ∼...

Nevertheless, the conformal group in each case is given by linear fractional transformations on the appropriate plane. Given a (Pseudo-)Riemannian manifold...

group to the quaternion context with linear fractional transformations. For the linear fractional transformations of an associative ring with 1, see projective...

mathematical optimization, fractional programming is a generalization of linear-fractional programming. The objective function in a fractional program is a ratio...