What does a Möbius transformation do?
These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. The Möbius transformations are the projective transformations of the complex projective line.
How do you get Möbius transformation?
Suppose the Möbius transformation is that M(z)=z+az+b, and then by M(−1)=i,M(i)=1+i, we can get that a=2+i,b=2−i, so the desired Möbius transformation is that M(z)=z+2+iz+2−i.
Is Inversion a Möbius transformation?
Inversion is NOT a Möbius transformation. In particular, inversion changes the orientation and Möbius transformations preserve it.
Is the identity map a Möbius transformation?
In particular, any Möbius transformation is determined by the images of three distinct points. (If there were two different transformations S, T having identical images at three distinct points, then S ◦ T−1 would have three fixed points and therefore be the identity map.)
Are Mobius transformation conformal?
It is also known as a bilinear transformation or a linear fractional transformation. Observation: Every Möbius transformation is a conformal map.
Are Mobius transformations Bijective?
Möbius Transformation is Bijection – ProofWiki.
Are Möbius transformation conformal?
Why Möbius transformation is conformal?
Any Möbius transformation which sends the real axis (union ∞) to itself and preserves the orientation of R will map the upper half-plane to itself conformally. Similarly any Möbius transformation which sends the (oriented) unit circle to itself will map the unit disc conformally to itself.
Are Mobius transformations orientation preserving?
All orientation-preserving isometries of are Mobius transformations, and all orientation-reversing isometries of are the composition of a Mobius transformation and reflection through the imaginary axis.
Is Möbius transformation conformal?
Every Möbius transformation is a conformal map. S (z) = (cz + d)a − c(az + b) (cz + d)2 = ad − bc (cz + d)2 = 0. Composition of two Möbius transformation is a Möbius transformation.
How many fixed points does a Mobius transformation have?
two fixed points
A Möbius transformation can have at most two fixed points unless it is an identity map. a−d (∞ if a = d) is the only fixed point of S . S(a) = T(a) = α, S(b) = T(b) = β and S(c) = T(c) = γ, where a, b, c are three distinct points in C∞.
How do you find the fixed points of Mobius transformation?
A Möbius transformation can have at most two fixed points unless it is an identity map. a−d (∞ if a = d) is the only fixed point of S . S(a) = T(a) = α, S(b) = T(b) = β and S(c) = T(c) = γ, where a, b, c are three distinct points in C∞.
How is the composition of two Mobius transformations constructed?
The composition of two Möbius transformations is again a Möbius transformation. Just as translations and rotations of the plane can be constructed from reflections across lines, the general Möbius transformation can be constructed from inversions about clines.
Is the Mobius transformation always a holomorphic function?
Thus a Möbius transformation is always a bijective holomorphic function from the Riemann sphere to the Riemann sphere. The set of all Möbius transformations forms a group under composition. This group can be given the structure of a complex manifold in such a way that composition and inversion are holomorphic maps.
Why do Mobius transformations take Clines to clines?
For instance, since inversion preserves clines, so do Möbius transformations, and since inversion preserves angle magnitudes, Möbius transformations preserve angles (as an even number of inversions). Theorem 3.4.5. Möbius transformations take clines to clines and preserves angles.
What are the Mobius transformations of the Riemann sphere?
The Möbius transformations are exactly the bijective conformal maps from the Riemann sphere to itself, i.e., the automorphisms of the Riemann sphere as a complex manifold; alternatively, they are the automorphisms of as an algebraic variety. Therefore, the set of all Möbius transformations forms a group under composition.