Title: Theory / Important / Math / Analysis / Complex / Visual+21 / 03 Mobius

Name: A Möbius transformation
Definition: A mapping of the form \(z\mapsto \frac{az+b}{cz+d}\) with \(a,b,c,d\in\mathbb C\)
Page: 122
Chapter: 3 I
Section: 1
Name2: linear(-fractional) transformation
Denoted by: \(M(z)\)
Name3: bilinear transformation
Name4: homographic transformation

Name: A singular Möbius mapping
Definition: A Möbius mapping \(\frac{az+b}{cz+d}\) with \(ad-bc=0\)
Page: 124
Chapter: 3 I
Section: 3
Kind: Ex

Name: A non-singular Möbius mapping
Definition: A Möbius mapping \(\frac{az+b}{cz+d}\) with \(ad-bc\ne0\)
Page: 124
Chapter: 3 I
Section: 3
Kind: Ex

Name: Complex inversion
Definition: \(z\mapsto\frac 1z\),
Page: 124
Chapter: 3 I
Section: 3
Kind: Ex
Definition2: \(re^{i\theta}\mapsto \frac 1re^{-i\theta}\): Symmetric angle and inverse of length

Question: The Möbius map \(z\mapsto\frac{az+b}{cz+d}\) can be decomposed as
Answer: \(z\mapsto z+\frac dc\)
Page: 124
Chapter: 3 I
Section: 1
Answer2: \(z\mapsto \frac 1z\)
Kind: Ex
Answer3: \(z\mapsto -\frac{ad-bc}{c^2}z\)
Answer4: \(z\mapsto z+\frac ac\)

Name: Geometric inversion
Definition: \(re^{i\theta}\mapsto \frac 1re^{i\theta}\)
Page: 125
Chapter: 3 II
Section: 1
Context: Planar geometry
Notation: \(\mathcal I_C\)
Intuition: Symmetry of \(z\) by the circle of radius 1

Name: Reflection in the circle \(K\)
Definition: \(z\mapsto\frac{R^2}{\bar z-\bar q}+q\)
Page: 125
Chapter: 3 II
Section: 1
Variables: \(q\in\mathbb C\),
\(R>0\)
Let \(K\) the circle of center \(q\) and radius \(R\)
Context: Planar geometry
Notation: \(\mathcal I_K\)
Definition2: [$]z\mapsto\frac{q\bar z+R^2-\abs{q}^2}{\bar z-\bar q}[/$]
Property: Involutory

Name: The reflection in the line \(L\)
Page: 37, 125
Chapter: 1IV, 3 II
Section: 3, 1
Variables: \(L\) a line
Notation: \(\mathfrak R_L\)
Property: Involutory

Name: An involutory mapping
Definition: A mapping \(f:X\to X\) such that \(f\circ f\) is the identity
Page: 125
Chapter: 3 II
Section: 1
Name2: A self-inverse mapping

Definition: [$]\abs{c-d}[/$]
Page: 126
Chapter: 3 II
Section: 1
Variables: \(c,d\in\mathbb C\)
Notation: \([cd]\)

Question: \([qa][q\mathcal I_K(a)] =\)
Answer: equals \(R^2\)
Page: 126
Chapter: 3 II
Section: 3
Variables: \(q\in\mathbb C\),
\(R>0\)
Let \(K\) the circle of center \(q\) and radius \(R\)
\(a\in\mathbb C\)

Question: The triangle \(q\mathcal I_K(a)\mathcal I_K(b)\) is
Answer: similar to the triangle \(qab\)
Page: 126
Chapter: 3 II
Section: 3
Variables: \(q\in\mathbb C\),
\(R>0\)
Let \(K\) the circle of center \(q\) and radius \(R\)
\(a,b\in\mathbb C\)

Question: \([\mathcal I_K(a)\mathcal I_K(b)]\)
Answer: equals \(\frac{R^2[ab]}{[qa][qb]}\)
Page: 126
Chapter: 3 II
Section: 3
Variables: \(q\in\mathbb C\),
\(R>0\)
\(K\) the circle of center \(q\) and radius \(R\)
\(a,b\in\mathbb C\)

Smaller0: \(L\) passes through the center of \(K\)
Greater0: \(\mathcal I_K(L)=L\)
Page: 127
Chapter: 3II
Section: 2
Variables: \(K\) a circle
\(L\) a line
Smaller to greater: iff

Smaller0: \(L\) does not pass through the center of \(K\)
Greater0: a circle passing through the center of \(K\)
Page: 127
Chapter: 3II
Section: 2
Variables: \(K\) a circle
\(L\) a line
Smaller to greater: iff
Prefix Greater: \(\mathcal I_K(L)\) is

Smaller0: \(C\) passes through the center of \(K\)
Greater0: a line not passing through the center of \(K\)
Page: 128
Chapter: 3II
Section: 2
Variables: \(K,C\) circles
\(L\) a line
Smaller to greater: iff
Prefix Greater: \(\mathcal I_K(C)\) is

Smaller0: \(C\) does not passes through the center of \(K\)
Greater0: a circle not passing through the center of \(K\)
Page: 128
Chapter: 3II
Section: 2
Variables: \(K,C\) circles
\(L\) a line
Smaller to greater: iff
Prefix Greater: \(\mathcal I_K(C)\) is

Smaller0: orthogonal to \(K\)
Greater0: \(C\)
Page: 129
Chapter: 3II
Section: 3
Variables: \(K,C\) circles
Index: (10)
Smaller to greater: iff
Prefix Greater: \(\mathcal I_K(C)\) is
Prefix Smaller: \(C\) is

Smaller0: \(C_1\) and \(C_2\) are orthogonal to \(K\)
Greater0: is the other intersection of \(C_1\cap C_2\)
Page: 129
Chapter: 3II
Section: 3
Variables: \(K,C_1,C_2\) circles
\(k\in C_1\cap C_2\)
Index: 10
Prefix Greater: \(\mathcal I_K(z)\)
Assuming: \(C_1\) intersects \(C_2\)

Smaller0: \(C_1\) and \(C_2\) are orthogonal to \(L\)
Greater0: is the other intersection of \(C_1\cap C_2\)
Page: 130
Chapter: 3II
Section: 3
Variables: \(L\) a line
\(C_1,C_2\) circles
\(k\in C_1\cap C_2\)
Index: 10
Prefix Greater: \(\mathcal I_K(z)\)
Assuming: \(C_1\) intersects \(C_2\)