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Conformal Mappings
Adrian Down
December 01, 2005
1 Angles between curves
1.1 Parameterization of lines
1.1.1 Motivation
Whenever we talk about line segments, we mean directed line segments. Ev- ery line segment must be specified with a direction. Our goal is to find an angle between two lines. We have to develop a logically consistent definition of this idea.
1.1.2 Parameterization
We can parameterize any line by
lj : t → z 0 + teıθj
By varying θj , a line in any direction can be obtained. Adding integer mul- tiples of 2π does not change the parametrization, thus any line can be pa- rameterized in infinitely many ways. The parameterization above is invariant under a number of transforma- tions:
- Multiplying the variable part of the parameterization by a real number does not change the direction of the line, only the speed at which it is traversed.
t → z 0 + 5teıθ It is most natural to define the line such that it is traversed at unit speed, so we take the multiplicative constant to be 1.
- It is possible to define the line with any nonzero complex number,
t → z 0 + t
w |w|
- It is also possible to shift the origin of the parameter without changing the line,
t → z 0 + (t − t 0 )eıθ
It is most natural to take t 0 = 0 for simplicity.
1.1.3 Angles
Definition (Angle between line segments). If θ is of the form θ 2 − θ 1 for some θ 2 , θ 1 ∈ R such that t → z 0 + teıθj^ parameterizes γj in the direction indicated, then θ is called an angle between γ 1 and γ 2.
Note. • The minus sign in the definition of θ is arbitrary. If it is switched consistently, all calculations will be identical.
- Keep in mind the following example to remember the sign of θ: the subtraction is taken so that if l 1 is the real axis, the angle to l 2 taken in the counter-clockwise direction is θ.
1.2 Tangent lines from a linear approximation
Assume
- γ : [a, b] → C is a path.
- t 0 ∈ (a, b)
- Define z 0 = γ(t 0 )
- γ′(t 0 ) exists
Then γ can be written as a linear approximation rear z 0 ,
γ(t) = z 0 + (t − t 0 )γ′(t) + r(t)
Note. It is possible that γ 1 and γ 2 go through the point at different times, thus the subscript on the t.
- The derivative at z 0 of both paths exists and neither is 0.
- θ ∈ R is an angle between γ 1 and γ 2 at z 0.
Then θ is an angle between their (directed) tangent lines at z 0
Note. This definition is somewhat ambiguous, since the paths could circle around and traverse z 0 again. We should really speak about the particular time at which the paths cross, but this is tedious to write, so we forgo the notation.
1.3.2 Example of poor choice of parameterization
Example.
γ 1 (s) = ıs γ 2 (t) = t^3
These paths parameterize the real and imaginary axes. It is clear that the paths meet at a right angle at the origin. However, our definition of the angle between the paths will not give a right angle, since the derivative of t^3 is 0 at t = 0. It is clear that this problem is a result of the parameterization. Taking γ 2 = t yields the correct result.
1.3.3 Angle is independent of parameterization
That the parameterization effects the angle between the paths is misleading. Suppose
- γ : [a, b] → C
- γ′(t 0 ) exists
- h : [c, d] → [a, b] and s 0 → t 0
Define a new path Γ as
Γ(s) = γ(h(s))
Note. Γ appears to be a new path, but it is really the old path re-parameterized.
With this definition,
Γ′(s 0 ) = γ′(h(s 0 )) · h′(s 0 ) = γ′(t 0 ) · h′(s 0 )
If the curve is re-parameterized, the direction of the tangent line does not change.
θ = any argument of Γ′(s 0 ) = any argument of γ′(t 0 ) · h′(s 0 ) = any argument of γ′(t 0 )
This shows that changing the parameterization of a curve does not change the angle of the tangent lines to the curve provided the transformation is a suitable one.
2 Conformal mappings
2.1 Motivation
Consider a function f from some domain D ⊂ C to C. We would like to establish conditions on f such that f preserves angles. Such a function is called a conformal mapping.
2.2 Definition
Definition (f preserves angles). f preserves angles at z 0 ∈ D if, for any paths γj with γj (tj ) = z 0 (γ j′ (tj ) exists and is nonzero), we have the paths
Γj (t) = f (γj (t))
Any angle between Γ 1 , Γ 2 at f (z 0 ) is an angle between γ 1 γ 2
Example. • Rotations preserves angles. For example,
f (z) = e
ıπ 4 · z
From the chain rule,
Γ′ 1 (t) = f ′(γ 1 (t 1 )) · γ′ 1 (t 1 ) = f ′(z 0 ) · γ 1 ′(t 1 ) Γ 2 (t) = f ′(z 0 ) · γ 2 ′(t 2 )
Let θ 1 , θ 2 be some arguments for γ 1 ′(t 1 ) and γ′(t 2 ), respectively. These quantities are nonzero by assumption. Let β be some argument for f ′(z 0 ). Then θ 1 + β is an argument for f ′(z 0 )γ 1 ′(t 1 ) = Γ′ 1 (t 1 ). We also have that θ 2 +β is an argument for f ′(z 0 )γ 2 ′(t 2 ) = Γ′ 2 (t 2 ). Referring to the definition for the angle between Γ′ 1 (t 1 ) and Γ′ 2 (t 2 ), call it θ,
θ = (θ 2 + β) − (θ 1 + β) = θ 2 − θ 1
is an angle between Γ 1 and Γ 2. θ 2 − θ 1 is the angle between γ 1 and γ 2. Thus the angle between γ 1 and γ 2 after the application of f is the same as that before.
Note. There is no need to talk about quotient groups to define angles, as is done in the text. Replace that discussion with the definitions given here.
3 Linear fractional transformations
3.1 Definition
Definition (LFT). A LFT is any mapping of the form
z →
az + b cz + d
where a, b, c, d ∈ C with ad − bc 6 = 0.
Note. It is no coincidence that the expression for a determinant appears in the condition. We will return to this later.
Example. • Some simple examples of LFTs are a scaling and a transla- tion,
z → az + b a 6 = 0
- Another canonical example is
z →
z As with any function, an LFT must be specified along with a domain. The general form of the LFT above is undefined for z = −dc if c 6 = 0. If c = 0, then we must have ad 6 = 0, which implies that it is possible to reduce the LFT to the form z → az + b.
3.2 Extended complex numbers
The most important properties of LFTs come from compositions thereof. The possibility that the input into one of the LFTs is not in its domain increases as the number of compositions is increased. In general, the domain decreases as the number of LFTs in a composition increases. To avoid this problem, we consider C with the addition of an extra point, usually denoted by ∞.
Definition (C). The extended complex numbers are defined as C with the addition of an extra point, usually called ∞,
C = C ∪ { ∞ }
Note. Geometrically, the best way to think about C is in terms of the Rie- mann sphere.
C is then the most natural domain on which to define LFTs.
Definition (LFT over C). For a general LFT,
f (z) =
az + b cz + d
- If c = 0, f (∞) = ∞.
- If c 6 = 0, f
−dc
lim |z|→∞
f (z) = f (∞) =
a c
Fact. Every such f is bijective C ↔ C
We can then read this off as
= L^0 @aa
′ (^) + bc′ (^) ab′ (^) + bd′ ca′^ + dc′^ cb′^ + dd′
1 A
= LM ·M ′ = LM ◦ LM ′
The composition of linear fractional transformations is expressible as a ma- trix. The matrix is given by the simple product of the two original M ma- trices.
3.3.3 Inverting LFTs
This also shows that every LFT is invertible. The inverse of any transfor- mation is that which when applied after the original transformation returns any coordinate to its initial position. Namely,
LM ◦ LM −^1 = I
Thus the matrix representing the inverse LFT is simply the matrix inverse of M. M is nonsingular by the assumption that ad − bc 6 = 0, so such an inverse always exists.
Note. ad − bc = 0 yields a constant function, which is not interesting.
f (z) =
az + b t(az + b)
t
This discussion also shows that the inverse of an LFT is also an LFT.
3.3.4 Group theory
The set of all LFTs is a group under the ground law of composition. It appears that we may have found a homomorphism between the set of LFTs and 2 by 2 matrices. However, the homomorphism fails, since two matrices can correspond to the same transformation.
z →
z + 10 z − ı
2 z + 20 2 z − ı
→
1 −ı
2 − 2 ı
