Linear fractional transformations are complex functions that map circles and lines to circles and lines. They're key players in complex analysis, offering a way to transform and analyze geometric shapes in the complex plane.
These transformations have cool properties like preserving angles and forming a group under composition. They're used in various fields, from physics to computer graphics, making them a versatile tool for solving complex problems.
Linear Fractional Transformations
Definition and Properties
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A linear fractional transformation (LFT) is a complex function of the form f(z)=(az+b)/(cz+d), where a, b, c, and d are complex numbers and ad−bc=0
LFTs are also known as Möbius transformations, homographic functions, or bilinear transformations
LFTs form a group under composition
The composition of two LFTs is another LFT
The inverse of an LFT is also an LFT
LFTs are conformal mappings
Preserve angles and the orientation of curves in the complex plane
The corresponds to the LFT with a=d=1 and b=c=0
The set of all LFTs is isomorphic to the projective special linear group PSL(2,C)
Fixed Points and Elementary Transformations
LFTs can be decomposed into a sequence of elementary transformations (translations, rotations, dilations, and inversions)
Translations correspond to LFTs of the form f(z)=z+b, where b is a complex number representing the displacement
Rotations and dilations are represented by LFTs of the form f(z)=az, where a is a non-zero complex number
The argument of a determines the rotation angle
The modulus of a determines the scaling factor
Inversions are represented by LFTs of the form f(z)=1/z
Map circles and lines to circles and lines, with the exception of lines through the origin, which are mapped to themselves
The of an LFT are the solutions to the equation f(z)=z
Can be found using the quadratic formula
LFTs can have at most two fixed points in the extended complex plane (including ∞)
Geometric Effects of LFTs
Mapping Circles and Lines
LFTs map circles and lines to circles and lines in the extended complex plane (including ∞)
To map a circle or line using an LFT
Apply the transformation to three or more points on the circle or line
Determine the image circle or line passing through the transformed points
LFTs preserve the of four distinct points in the extended complex plane
Can be used to solve problems involving the mapping of specific points or regions
The pre-image of a circle or line under an LFT can be found by applying the to the image circle or line
Mapping Regions and Applications
LFTs can be used to map the upper half-plane, the unit disk, or other regions in the complex plane to more convenient domains for analysis or computation
Applications of LFTs in various fields
Physics (conformal field theory)
Engineering (signal processing)
Computer graphics (image transformations)
Mapping with LFTs
Composition of LFTs
The composition of two LFTs, denoted by (f∘g)(z), is another LFT obtained by substituting g(z) for z in the expression for f(z) and simplifying the result
The set of all LFTs forms a group under composition
The identity transformation as the identity element
The inverse of an LFT as the group inverse
Möbius Transformations and Classification
Möbius transformations are equivalent to LFTs and are often studied in the context of hyperbolic geometry and complex analysis
The group of Möbius transformations is isomorphic to the projective special linear group PSL(2,C)
The quotient group of the special linear group SL(2,C) by its center ±I
Classification of Möbius transformations based on their fixed points and trace
Parabolic transformations (one fixed point)
Elliptic transformations (two fixed points, trace is real and ∣trace∣<2)
Hyperbolic transformations (two fixed points, trace is real and ∣trace∣>2)
Loxodromic transformations (two fixed points, trace is complex)
LFT Composition and Möbius Transformations
Composition and Group Structure
The composition of LFTs is associative and forms a group
The identity LFT is f(z)=z
The inverse of an LFT f(z)=(az+b)/(cz+d) is f−1(z)=(dz−b)/(−cz+a)
The group of LFTs is non-abelian, meaning that the order of composition matters
In general, (f∘g)(z)=(g∘f)(z)
The group of LFTs acts on the extended complex plane (including ∞) by permuting points according to the transformation
Relation to Other Areas of Mathematics
Möbius transformations are closely related to projective geometry
The extended complex plane can be identified with the complex projective line CP1
LFTs correspond to projective transformations of CP1
The group of Möbius transformations is isomorphic to the group of isometries of the hyperbolic plane
The upper half-plane model of hyperbolic geometry
The Poincaré disk model of hyperbolic geometry
Möbius transformations have applications in the study of rational functions and algebraic curves
Rational functions can be expressed as the composition of LFTs and polynomials
LFTs can be used to simplify and analyze the geometry of algebraic curves