Introduction to Möbius Transformations

A Möbius transformation is a function of the form $w=\frac{az+b}{cz+d}$ where $a$, $b$, $c$, and $d$ are complex constants and $z$ is a complex variable. These transformations are fundamental in complex analysis because they map the extended complex plane to itself, preserving angles and sending circles and lines to circles or lines. When $ad-bc\ne 0$, the transformation is invertible, and its inverse is also a Möbius transformation.

One way to appreciate Möbius transformations is through the cross-ratio. Given four distinct complex numbers $z$₁, $z$₂, $z$₃, and $z$₄, the cross-ratio is defined as $\frac{z}{}$₁−z₃z₂−z₄z₁−z₄z₂−z₃. Möbius transformations are exactly the functions that preserve this quantity. This property plays a key role in projective geometry and conformal mapping theory.

Because Möbius transformations are conformal, they preserve angles locally. When you apply such a transformation to a grid, small shapes retain their form but may be scaled or rotated. A special case is the mapping $z\to \frac{1}{z}$, which sends large circles outside the unit disk to small circles inside it. Combining reflections, translations, dilations, and inversions, you can build any Möbius transformation. This illustrates how flexible these mappings are for reconfiguring geometric domains.

Another important perspective comes from linear fractional transformations. If you represent complex numbers as vectors in $C^2$, a Möbius transformation corresponds to multiplying by a 2×2 complex matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$. The projective action of this matrix on homogeneous coordinates yields the transformation formula. Thus, Möbius transformations form a group under composition, often denoted $\mathrm{PSL}\left(2C\right)$.

Möbius transformations have elegant geometric interpretations. They map generalized circles—meaning both circles and lines—to other generalized circles. The mapping may interchange interior and exterior regions, or send a circle to a line when the denominator vanishes at a boundary point. In the hyperbolic plane model known as the Poincaré disk, Möbius transformations act as isometries. Consequently, they provide a powerful toolbox for visualizing non-Euclidean geometry.

This calculator lets you experiment with Möbius transformations numerically. Input complex coefficients using the form a+bi. For instance, entering 1+i means $a=1+i$. After specifying a complex number $z$, pressing Transform computes $w$ via $\frac{az+b}{cz+d}$. The script parses the coefficients with math.js so you can use standard complex notation.

Properties and Applications

One remarkable property is that Möbius transformations preserve circles orthogonally. If a circle passes through the point at infinity—effectively a straight line—it will remain a line after transformation. Otherwise, the image is again a circle. This is straightforward to verify algebraically, and it hints at why these transformations are so useful in complex analysis: they convert complicated domains into shapes that may be easier to study.

Consider the mapping $z\to \frac{z}{+}z+2$. It sends the left half-plane to the unit disk. Such mappings allow you to transform boundary-value problems on one domain to another domain where the solution is known or simpler. Engineers working with filter design sometimes exploit these properties to transform stability regions between the complex plane and the unit disk.

In projective geometry, Möbius transformations help classify how lines and circles intersect. Since they preserve cross-ratios, they maintain the harmonic division of four collinear points. This invariance underpins constructions in classical geometry and also appears in modern graphics algorithms that require conformal mapping of textures.

There is also a connection to number theory. The group $\mathrm{PSL}\left(2Z\right)$—Möbius transformations with integer coefficients and determinant one—acts on the upper half-plane, tessellating it into fundamental regions. This action is central to the theory of modular forms and elliptic functions. Consequently, studying Möbius transformations sheds light on deep arithmetic properties.

When applied iteratively, Möbius transformations can generate fractal-like structures known as limit sets. By varying the coefficients, you produce patterns reminiscent of the well-known Apollonian gasket. These visualizations highlight the rich dynamical behavior hidden in what might appear to be a simple rational function.

Finally, Möbius transformations have practical uses in cartography and computer vision. Stereographic projection, which maps a sphere to a plane, is itself a Möbius transformation. Similarly, some camera lens distortions can be approximated by such mappings, and understanding them allows for easier correction in post-processing.

The calculator below is straightforward to use. Simply enter values for $a$, $b$, $c$, and $d$. Then specify a complex point $z$. The script ensures that $ad-bc\ne 0$ before computing the result. If the denominator $cz+d$ happens to be zero, the image is considered to be at infinity, represented by the text Infinity. Experiment with varying coefficients to observe how lines, circles, and angles transform under this versatile class of functions.