8.2 Möbius transformations

Möbius transformations are powerful tools in complex analysis, mapping circles and lines to circles and lines. They're defined as complex functions of the form (az + b) / (cz + d), where a, b, c, d are constants and ad - bc ≠ 0.

These transformations form a group under composition and are uniquely determined by their action on any three distinct points. They can be classified as elliptic, parabolic, or hyperbolic based on their parameters, each type having distinct geometric behaviors and fixed point properties.

Möbius Transformations: General Form and Properties

Definition and Basic Properties

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• A Möbius transformation is a complex function of the form $f(z) = (az + b) / (cz + d)$, where $a$, $b$, $c$, $d$ are complex constants and $ad - bc ≠ 0$
  • The condition $ad - bc ≠ 0$ ensures that the transformation is invertible and well-defined
  • Example: $f(z) = (2z + 1) / (z - 3)$ is a Möbius transformation with $a = 2$, $b = 1$, $c = 1$, and $d = -3$
• Möbius transformations are conformal mappings, preserving angles and shapes of infinitesimal figures in the complex plane
  • Conformality implies that Möbius transformations preserve the local geometry of the complex plane
  • Example: If two curves intersect at a right angle, their images under a Möbius transformation will also intersect at a right angle

Group Properties and Mapping Characteristics

• Möbius transformations form a group under composition, meaning the composition of two Möbius transformations is another Möbius transformation, and the inverse of a Möbius transformation is also a Möbius transformation
  • The group of Möbius transformations is called the Möbius group or the linear fractional group
  • Example: If $f(z) = (2z + 1) / (z - 3)$ and $g(z) = (z + 2) / (3z - 1)$, then $(f ∘ g)(z)$ and $(g ∘ f)(z)$ are also Möbius transformations
• Möbius transformations map circles and lines to circles and lines in the complex plane, including the extended complex plane (the complex plane with the point at infinity)
  • Lines can be considered as circles passing through the point at infinity
  • Example: The Möbius transformation $f(z) = 1 / z$ maps the unit circle $|z| = 1$ to the real line plus the point at infinity
• Möbius transformations are uniquely determined by their action on any three distinct points in the extended complex plane
  • Given any three distinct points $z₁$, $z₂$, and $z₃$ in the extended complex plane and any three distinct points $w₁$, $w₂$, and $w₃$, there exists a unique Möbius transformation that maps $z₁$ to $w₁$, $z₂$ to $w₂$, and $z₃$ to $w₃$
  • Example: The Möbius transformation that maps $0$ to $1$, $1$ to $i$, and $∞$ to $-1$ is $f(z) = (z - i) / (z + i)$

Möbius Transformations: Types and Parameters

Classification by Parameters

• A Möbius transformation is called a linear fractional transformation (LFT) if $c ≠ 0$ and a homographic transformation if $c = 0$
  • LFTs have a more general form and can represent a wider range of transformations
  • Homographic transformations are a special case of LFTs and include linear transformations and translations
  • Example: $f(z) = (2z + 1) / (z - 3)$ is an LFT, while $g(z) = 2z + 1$ is a homographic transformation
• A Möbius transformation is called elliptic if $|a + d| < 2$, parabolic if $|a + d| = 2$, and hyperbolic if $|a + d| > 2$, where $a$ and $d$ are the coefficients of the transformation
  • The type of a Möbius transformation determines its geometric behavior and fixed point properties
  • Example: $f(z) = (i/2)z$ is elliptic, $g(z) = z + 1$ is parabolic, and $h(z) = 2z$ is hyperbolic

Geometric Interpretation of Types

• Elliptic transformations have two fixed points in the complex plane and no fixed points at infinity, representing a rotation around a fixed point
  • The two fixed points are conjugate to each other and lie inside the unit circle
  • Example: $f(z) = (i/2)z$ represents a rotation by 90° around the origin
• Parabolic transformations have one fixed point in the extended complex plane, representing a translation or a rotation combined with a dilation
  • The fixed point can be either in the complex plane or at infinity
  • Example: $g(z) = z + 1$ represents a translation by 1 unit to the right
• Hyperbolic transformations have two fixed points, one in the complex plane and one at infinity, representing a dilation or a combination of translation and dilation
  • The fixed points are real and lie on opposite sides of the unit circle
  • Example: $h(z) = 2z$ represents a dilation by a factor of 2 centered at the origin

Special Cases

• Special cases of Möbius transformations include the identity transformation ($a = d = 1$, $b = c = 0$), pure rotations ($a = d$, $b = -c̄$, $|a|² + |b|² = 1$), and pure dilations ($a = d$, $b = c = 0$, $|a| ≠ 1$)
  • The identity transformation maps every point to itself
  • Pure rotations preserve distances and rotate the complex plane around a fixed point
  • Pure dilations scale the complex plane by a constant factor centered at a fixed point
  • Example: $f(z) = z$ is the identity transformation, $g(z) = (1/√2)(z - i)$ is a pure rotation by 45°, and $h(z) = 3z$ is a pure dilation by a factor of 3

Composing and Decomposing Möbius Transformations

Composition of Möbius Transformations

• The composition of two Möbius transformations $f(z) = (a₁z + b₁) / (c₁z + d₁)$ and $g(z) = (a₂z + b₂) / (c₂z + d₂)$ is another Möbius transformation $h(z) = (az + b) / (cz + d)$, where $a = a₁a₂ + b₁c₂$, $b = a₁b₂ + b₁d₂$, $c = c₁a₂ + d₁c₂$, and $d = c₁b₂ + d₁d₂$
  • The coefficients of the composed transformation can be found by multiplying the matrices representing the individual transformations
  • Example: If $f(z) = (2z + 1) / (z - 3)$ and $g(z) = (z + 2) / (3z - 1)$, then $(f ∘ g)(z) = (5z + 11) / (3z + 5)$
• The order of composition matters for Möbius transformations, as they are generally not commutative
  • In general, $(f ∘ g)(z) ≠ (g ∘ f)(z)$ for Möbius transformations $f$ and $g$
  • Example: For $f(z) = 1 / z$ and $g(z) = z + 1$, $(f ∘ g)(z) = 1 / (z + 1)$, while $(g ∘ f)(z) = 1 / z + 1$

Decomposition of Möbius Transformations

• Any Möbius transformation can be decomposed into a sequence of simpler transformations, such as translations, rotations, dilations, and inversions
  • Translations, rotations, and dilations are special cases of Möbius transformations, while inversions are Möbius transformations of the form $f(z) = 1 / z$
  • Example: The Möbius transformation $f(z) = (2z + 1) / (z - 3)$ can be decomposed into a translation by 3, followed by an inversion, a dilation by 2, another translation by 1/2, and finally another inversion
• The decomposition of a Möbius transformation can help analyze its geometric effect on the complex plane and understand its fixed points and critical points
  • Each component of the decomposition has a specific geometric interpretation, and the overall effect can be understood by combining these interpretations
  • Example: The decomposition of $f(z) = (2z + 1) / (z - 3)$ reveals that it maps the point at infinity to -1/2, the point 3 to infinity, and has a critical point at z = 1

Fixed Points and Critical Points of Möbius Transformations

Fixed Points

• A fixed point of a Möbius transformation $f(z)$ is a complex number $z₀$ such that $f(z₀) = z₀$. Fixed points can be found by solving the equation $az + b = z(cz + d)$ for $z$
  • The fixed points are the roots of the quadratic equation $cz² + (d - a)z - b = 0$
  • Example: For $f(z) = (2z + 1) / (z - 3)$, the fixed points are $z = 2 ± √5$
• A Möbius transformation has either one or two fixed points in the extended complex plane, depending on its type (elliptic, parabolic, or hyperbolic)
  • Elliptic transformations have two fixed points in the complex plane, parabolic transformations have one fixed point in the extended complex plane, and hyperbolic transformations have one fixed point in the complex plane and one at infinity
  • Example: The elliptic transformation $f(z) = (i/2)z$ has fixed points at $0$ and $∞$, the parabolic transformation $g(z) = z + 1$ has a fixed point at $∞$, and the hyperbolic transformation $h(z) = 2z$ has fixed points at $0$ and $∞$
• The fixed points of a Möbius transformation can be classified as attractive, repulsive, or indifferent based on the behavior of nearby points under iteration of the transformation
  • An attractive fixed point "attracts" nearby points under iteration, a repulsive fixed point "repels" nearby points, and an indifferent fixed point has a neutral behavior
  • Example: For the hyperbolic transformation $h(z) = 2z$, $0$ is an attractive fixed point, and $∞$ is a repulsive fixed point

Critical Points

• A critical point of a Möbius transformation $f(z)$ is a complex number $z₀$ such that $f'(z₀) = 0$. The derivative of a Möbius transformation is given by $f'(z) = (ad - bc) / (cz + d)²$
  • Critical points occur where the denominator of the derivative vanishes, i.e., where $cz + d = 0$
  • Example: For $f(z) = (2z + 1) / (z - 3)$, the critical point is $z = 3$
• Critical points of a Möbius transformation correspond to the points where the transformation has the greatest local distortion or where the mapping is not conformal
  • At a critical point, angles and shapes are not preserved, and the transformation may map infinitesimal circles to infinitesimal ellipses
  • Example: The critical point $z = 3$ of $f(z) = (2z + 1) / (z - 3)$ is mapped to infinity, indicating a significant distortion near this point
• The number and location of critical points provide information about the geometric properties of the Möbius transformation and its effect on the complex plane
  • The number of critical points is related to the type of the transformation (elliptic, parabolic, or hyperbolic) and its fixed point structure
  • Example: Elliptic transformations have no critical points in the complex plane, parabolic transformations have one critical point, and hyperbolic transformations have two critical points (one in the complex plane and one at infinity)