Conformal and Bilinear Transformations

Introduction to Transformations

In mathematics, a transformation is a function that maps a set of points to another set of points. In the context of complex analysis, we are primarily interested in transformations of the complex plane. These transformations can alter the geometric properties of shapes and regions.

A transformation can be thought of as a way to "move" or "reshape" points from one plane to another. For instance, a simple translation shifts every point by a fixed amount, while a rotation spins points around a fixed center. Complex analysis allows us to study these transformations using the powerful tools of complex numbers and functions.

What is a Bilinear Transformation?

A bilinear transformation, also known as a Möbius transformation or fractional linear transformation, is a function of the form:

$$w = f(z) = \frac{az + b}{cz + d}$$

where $a, b, c, d$ are complex constants and $ad - bc \neq 0$. The condition $ad - bc \neq 0$ ensures that the transformation is indeed non-trivial and invertible.

These transformations are fundamental in complex analysis because they map circles and lines in the complex plane to circles and lines. A "generalized circle" refers to either a circle or a line. This property makes them extremely useful for solving geometric problems.

The transformation can be extended to the extended complex plane (the complex plane plus a point at infinity, denoted by $\infty$). If $c \neq 0$, then $f(-d/c) = \infty$ and $f(\infty) = a/c$. If $c = 0$, then $f(\infty) = \infty$.

Properties of Bilinear Transformations

Bilinear transformations possess several key properties that make them important:

• They map generalized circles to generalized circles. This is arguably their most significant property. A circle can be mapped to another circle, a circle to a line, a line to a circle, or a line to another line.
• They are conformal (except at poles). We will discuss conformality in detail later, but it means they preserve angles locally.
• They form a group under composition. If $f(z)$ and $g(z)$ are bilinear transformations, then their composition $f(g(z))$ is also a bilinear transformation. The set of all bilinear transformations forms a group under the operation of composition.
• They are invertible. For every bilinear transformation $w = f(z)$, there exists an inverse transformation $z = f^{-1}(w)$ which is also bilinear.
• A bilinear transformation is uniquely determined by its action on any three distinct points. Given three distinct points $z_1, z_2, z_3$ and their images $w_1, w_2, w_3$ (where $w_i$ are distinct), there is a unique bilinear transformation $w = f(z)$ such that $f(z_i) = w_i$ for $i=1, 2, 3$.

The Cross-Ratio

The cross-ratio of four distinct points $z_1, z_2, z_3, z_4$ is defined as:

$$(z_1, z_2; z_3, z_4) = \frac{(z_1 - z_3)(z_2 - z_4)}{(z_1 - z_4)(z_2 - z_3)}$$

A fundamental property of bilinear transformations is that they preserve the cross-ratio. If $w = f(z)$ is a bilinear transformation, and $z_1, z_2, z_3, z_4$ are distinct points, and $w_i = f(z_i)$ for $i=1, 2, 3, 4$, then:

$$(z_1, z_2; z_3, z_4) = (w_1, w_2; w_3, w_4)$$

This property can be used to find the bilinear transformation that maps three given points to three other given points. If we want to find $w=f(z)$ such that $f(z_1)=w_1$, $f(z_2)=w_2$, and $f(z_3)=w_3$, we can use the cross-ratio formula:

$$\frac{(z, z_1; z_2, z_3)}{(w, w_1; w_2, w_3)} = 1$$

Or, more commonly written as:

$$\frac{(z - z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)} = \frac{(w - w_1)(w_2 - w_3)}{(w - w_3)(w_2 - w_1)}$$

This equation can be rearranged to solve for $w$ in terms of $z$, yielding the specific bilinear transformation.

Examples of Bilinear Transformations

Let's look at some common examples:

• Translation: $w = z + b$ (here $a=1, c=0, d=1$). This shifts every point by $b$.
• Rotation: $w = az$ where $|a|=1$ (here $b=0, c=0, d=1$). This rotates points around the origin.
• Magnification/Contraction: $w = az$ where $a$ is real and positive (here $b=0, c=0, d=1$). This scales distances from the origin.
• Inversion: $w = 1/z$ (here $a=0, b=1, c=1, d=0$). This is a powerful transformation that maps circles/lines to circles/lines and is crucial in understanding Möbius transformations.

Consider the transformation $w = \frac{z+1}{z-1}$. Here $a=1, b=1, c=1, d=-1$. $ad-bc = (1)(-1) - (1)(1) = -1 - 1 = -2 \neq 0$. Let's see where it maps some points: $z=0 \implies w = \frac{0+1}{0-1} = -1$. $z=1 \implies w = \infty$. $z=-1 \implies w = \frac{-1+1}{-1-1} = \frac{0}{-2} = 0$. $z=2 \implies w = \frac{2+1}{2-1} = \frac{3}{1} = 3$. $z=\infty \implies w = a/c = 1/1 = 1$.

This transformation maps the imaginary axis ($z = iy$) to the unit circle $|w|=1$. Let $z = iy$. Then $w = \frac{iy+1}{iy-1}$. $|w| = \left|\frac{1+iy}{-1+iy}\right| = \frac{|1+iy|}{|-1+iy|} = \frac{\sqrt{1^2+y^2}}{\sqrt{(-1)^2+y^2}} = \frac{\sqrt{1+y^2}}{\sqrt{1+y^2}} = 1$. So, the imaginary axis (a line) is mapped to the unit circle (a circle).

What is a Conformal Transformation?

A transformation $w = f(z)$ is called conformal at a point $z_0$ if it preserves angles between curves passing through $z_0$. More precisely, if two smooth curves intersect at $z_0$ with an angle $\theta$, their images under $f$ will intersect at $f(z_0)$ with the same angle $\theta$.

A transformation is conformal in a region if it is conformal at every point in that region.

The fundamental theorem regarding conformality states:

A complex analytic function $f(z)$ is conformal at all points $z_0$ where $f'(z_0) \neq 0$.

This means that if a function is differentiable and its derivative is non-zero at a point, the transformation represented by that function will preserve angles at that point. This is a very strong geometric property.

Properties of Conformal Transformations

Conformal transformations have several important geometric properties:

• Angle Preservation: As defined, they preserve the magnitude and orientation of angles between intersecting curves.
• Local Scaling: Near a point $z_0$ where $f'(z_0) \neq 0$, the transformation acts like a local rotation and magnification. The magnification factor is $|f'(z_0)|$. The angle of rotation is $\arg(f'(z_0))$.
• Mapping of Small Shapes: Small shapes are mapped to approximately similar shapes, with a scaling factor of $|f'(z)|$ and a rotation of $\arg(f'(z))$.
• Analyticity is Key: Conformality is deeply tied to analyticity. If a function is not analytic, it might not preserve angles consistently.

Relationship between Bilinear and Conformal Transformations

Bilinear transformations ($w = \frac{az+b}{cz+d}$) are a special class of conformal transformations.

Let's compute the derivative of $w = f(z) = \frac{az+b}{cz+d}$:

Using the quotient rule:

$$f'(z) = \frac{a(cz+d) - c(az+b)}{(cz+d)^2} = \frac{acz + ad - acz - bc}{(cz+d)^2} = \frac{ad-bc}{(cz+d)^2}$$

Since $ad-bc \neq 0$, the derivative $f'(z)$ is non-zero everywhere except at $z = -d/c$, which is the pole of the transformation. Therefore, any bilinear transformation is conformal at all points $z$ in the complex plane where $z \neq -d/c$.

At the pole $z = -d/c$, the transformation maps this point to infinity. While angles are not preserved in the traditional sense at this point, the behavior is well-understood and consistent with the extension to the Riemann sphere.

So, all bilinear transformations are conformal, except at their single pole.

Conformal Mapping Theorem

One of the most powerful results in complex analysis is the Riemann Mapping Theorem. It states that if $D$ is any simply connected domain in the complex plane (meaning it has no "holes") and $D$ is not the entire complex plane itself, then there exists a conformal mapping (a one-to-one, onto, conformal function) from $D$ onto the unit disk $\{w \in \mathbb{C} : |w| < 1\}$. Furthermore, this mapping is unique if we specify the image of one point and the argument of the derivative at that point.

This theorem is profound because it implies that all "nice" open regions in the complex plane are conformally equivalent to the unit disk. This means we can transform complicated regions into a simple disk, solve problems on the disk (where it's often easier), and then map the solution back to the original region.

Applications of Conformal and Bilinear Transformations

These transformations have wide-ranging applications in various fields:

• Fluid Dynamics: Used to model fluid flow around obstacles. For example, mapping the flow around a cylinder to the flow around a flat plate.
• Electrostatics: Solving problems involving electric potentials in 2D. Conformal mappings can transform complex boundary shapes into simpler ones, making potential calculations feasible.
• Heat Conduction: Analyzing steady-state temperature distributions in regions with complex geometries.
• Cartography: Map projections are conformal transformations that preserve angles, ensuring that directions are represented accurately on the map, although areas and distances may be distorted. The Mercator projection is a famous example.
• Image Processing and Computer Graphics: Used for image warping and manipulation.
• Aerodynamics: Designing airfoils by transforming simpler shapes.

Example: Mapping the Upper Half-Plane to the Unit Disk

Let's consider a specific example of a conformal mapping. We want to map the upper half-plane $\text{Im}(z) > 0$ to the unit disk $|w| < 1$.

A common transformation used for this is:

$$w = f(z) = \frac{z - i}{z + i}$$

Let's check if this is a bilinear transformation. Here $a=1, b=-i, c=1, d=i$. $ad - bc = (1)(i) - (-i)(1) = i - (-i) = 2i \neq 0$. So it is a bilinear transformation.

Let's see where it maps the real axis ($y=0$ or $z$ is real). If $z$ is real, $z = x$. $w = \frac{x-i}{x+i}$. $|w| = \left|\frac{x-i}{x+i}\right| = \frac{|x-i|}{|x+i|} = \frac{\sqrt{x^2 + (-1)^2}}{\sqrt{x^2 + 1^2}} = \frac{\sqrt{x^2+1}}{\sqrt{x^2+1}} = 1$. So, the real axis (boundary of the upper half-plane) is mapped to the unit circle $|w|=1$.

Now let's check a point in the upper half-plane, say $z=i$. $w = f(i) = \frac{i - i}{i + i} = \frac{0}{2i} = 0$. The point $z=i$ is mapped to the center of the unit disk, $w=0$.

Since the transformation maps the boundary of the upper half-plane to the boundary of the unit disk, and maps a point inside the upper half-plane to the center of the unit disk, it must map the entire upper half-plane to the interior of the unit disk.

The derivative is $f'(z) = \frac{(1)(z+i) - (1)(z-i)}{(z+i)^2} = \frac{z+i-z+i}{(z+i)^2} = \frac{2i}{(z+i)^2}$. For $z$ in the upper half-plane, $z+i \neq 0$, so $f'(z) \neq 0$. This confirms that the mapping is conformal in the upper half-plane.

Example: Mapping the Unit Disk to Itself

Consider mapping the unit disk $|z| < 1$ to itself. A general bilinear transformation that maps the unit disk to itself is of the form:

$$w = f(z) = e^{i\alpha} \frac{z - z_0}{1 - \bar{z_0} z}$$

where $\alpha$ is a real number and $z_0$ is any point inside the unit disk ($|z_0| < 1$).

Let's verify this. If $|z|=1$, then $z\bar{z}=1$, so $\bar{z} = 1/z$. $|w|^2 = w\bar{w} = \left(e^{i\alpha} \frac{z - z_0}{1 - \bar{z_0} z}\right) \overline{\left(e^{i\alpha} \frac{z - z_0}{1 - \bar{z_0} z}\right)}$ $|w|^2 = |e^{i\alpha}|^2 \left|\frac{z - z_0}{1 - \bar{z_0} z}\right|^2 = 1 \cdot \frac{(z - z_0)(\bar{z} - \bar{z_0})}{(1 - \bar{z_0} z)(1 - z_0 \bar{z})}$ $|w|^2 = \frac{z\bar{z} - z\bar{z_0} - \bar{z}z_0 + z_0\bar{z_0}}{1 - z\bar{z_0} - \bar{z}z_0 + z_0\bar{z_0}z\bar{z}}$ Substitute $|z|=1$ (so $z\bar{z}=1$ and $\bar{z}=1/z$): $|w|^2 = \frac{1 - z\bar{z_0} - (1/z)z_0 + |z_0|^2}{1 - z\bar{z_0} - (1/z)z_0 + |z_0|^2}$ The numerator and denominator are identical, so $|w|^2 = 1$. This means points on the unit circle are mapped to the unit circle.

Now consider $z=0$. $w = e^{i\alpha} \frac{0 - z_0}{1 - \bar{z_0} \cdot 0} = e^{i\alpha} \frac{-z_0}{1} = -e^{i\alpha}z_0$. The modulus is $|w| = |-e^{i\alpha}z_0| = |e^{i\alpha}||-z_0| = 1 \cdot |z_0| = |z_0|$. If $|z_0| < 1$, then $|w| < 1$. So, the center of the disk is mapped to a point inside the disk. This implies the entire unit disk is mapped to the unit disk.

This type of transformation is crucial for proving the Riemann Mapping Theorem and for understanding the automorphism group of the unit disk.

Summary of Key Concepts

Bilinear Transformation: $w = \frac{az+b}{cz+d}$ with $ad-bc \neq 0$. Maps generalized circles to generalized circles. Forms a group. Preserves cross-ratio.

Conformal Transformation: A transformation that preserves angles locally. If $f(z)$ is analytic and $f'(z_0) \neq 0$, then $f$ is conformal at $z_0$.

Relationship: Bilinear transformations are conformal everywhere except at their pole.

Riemann Mapping Theorem: Any non-trivial simply connected domain can be conformally mapped onto the unit disk.

Applications: Essential tools in physics and engineering for solving boundary value problems in regions with complex geometries.