2.3 Linear fractional transformations

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

• 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 identity transformation 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 fixed points 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 cross-ratio 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 inverse transformation 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 $ℂℙ^1$
  • LFTs correspond to projective transformations of $ℂℙ^1$
• 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