Möbius Transformations
Maps of the form (az+b)/(cz+d).
Key idea
Maps of the form (az+b)/(cz+d).
In depth
f(z) = (az + b)/(cz + d) with ad − bc ≠ 0. They map circles and lines to circles and lines, form a group, and are determined by images of three points. They are conformal automorphisms of the Riemann sphere.
Worked example
f(z) = 1/z maps the unit circle to itself and swaps inside and outside.
Formula
Real-world uses
- Designing lenses and mapping shapes on the sphere in graphics and physics
Common mistake
Forgetting the point at infinity.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Möbius Transformations?
Maps of the form (az+b)/(cz+d). It is a topic at the expert level in Analysis & Topology.
What is the formula for Möbius Transformations?
The key formula is (az+b)/(cz+d).
Where is Möbius Transformations used in real life?
Real-world examples: Designing lenses and mapping shapes on the sphere in graphics and physics.
What should I learn before Möbius Transformations?
Make sure you are comfortable with Conformal Maps, the previous topic in Analysis & Topology.
What is a common mistake with Möbius Transformations?
Forgetting the point at infinity.
How can I practise Möbius Transformations?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Möbius Transformations?
Move on to Riemann Mapping Theorem next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.