Möbius Transformation Calculator
Introduction
A Möbius transformation looks compact on paper, but it can produce surprisingly rich geometry. With one formula, you can combine a translation, a rotation, a scaling, and an inversion into a single map on the complex plane. This calculator is designed to make that process practical: enter the four complex coefficients a, b, c, and d, choose an input point z, and the tool computes the transformed point w. If the denominator vanishes, the page reports that the image is at infinity, which is exactly the special behavior that makes Möbius maps so distinctive.
If you are using this tool while studying complex analysis, conformal mapping, or hyperbolic geometry, it helps to think of the calculator as both an algebra checker and an intuition builder. Algebraically, it substitutes your values into a linear fractional expression and simplifies the result. Geometrically, it tells you where a point moves under a map that preserves angles and sends lines and circles to other lines or circles. There are usually no physical units attached to these numbers; they are complex coordinates. In an applied setting, if your complex values do stand for measured quantities, then all terms in the numerator and denominator should be dimensionally consistent so the ratio is meaningful.
The sections below explain the formula, the meaning of each input, the determinant condition that keeps the transformation non-degenerate, and how to interpret special cases such as poles and inversions. A worked example is included, and the optional mini-game at the end lets you feel the mapping dynamically by steering points on the input plane and watching their images move on the output plane.
What Is a Möbius Transformation?
A Möbius transformation, also called a linear fractional transformation, is a complex function of the form
w = f(z) = .
Here a, b, c, d, and z are complex numbers. The output w is another complex number unless the denominator becomes zero, in which case the image is the point at infinity on the extended complex plane. That extended plane is often visualized as the Riemann sphere, which is why Möbius maps are so important: they naturally describe transformations on a sphere-like compactification of the complex plane.
One reason these maps appear so often is that they are flexible without being chaotic. If c = 0, the formula reduces to an affine map w = az + b, which means a rotation, scaling, and translation. When c ≠ 0, the denominator introduces inversion-like behavior, creating a pole at the point where cz + d = 0. Near that pole, the image can move very rapidly, which is why Möbius transformations are excellent examples for understanding singularities and conformal structure.
Geometrically, these maps are angle-preserving wherever the derivative is non-zero, and they send generalized circles to generalized circles. In this context, a generalized circle means either an ordinary circle or a straight line. So even though the algebra looks like a ratio, the geometry is highly structured rather than arbitrary.
Möbius Transformation Formula
The general formula used by this calculator is
w = .
The transformation is well-defined whenever the denominator c·z + d is not zero. If c·z + d = 0, the image of z is the point at infinity. In a numerical calculator, that special value is usually reported with a text label rather than with ordinary real and imaginary parts.
A fundamental non-degeneracy condition is that a·d − b·c ≠ 0. This determinant is what prevents the formula from collapsing into a constant or otherwise degenerate mapping. When the determinant is non-zero, the transformation is invertible, and the inverse is again a Möbius transformation. That is why textbooks often treat ad − bc ≠ 0 as part of the definition rather than just a side note.
MathML version of the formula
The core formula can also be written in MathML for better machine readability:
Cross-ratio viewpoint
Möbius transformations are exactly the maps that preserve the cross-ratio of four distinct complex numbers. For four points z1, z2, z3, and z4, the cross-ratio is
((z₁ − z₃)(z₂ − z₄)) / ((z₁ − z₄)(z₂ − z₃)).
This point of view matters because it explains why Möbius transformations belong as much to projective geometry as to complex analysis. If you know that a map preserves cross-ratio, you know you are looking at a Möbius transformation. If you know the images of three points under a non-degenerate Möbius map, you can usually determine the transformation uniquely.
Geometric Properties and Interpretation
Möbius transformations have several key geometric properties that make them useful far beyond routine arithmetic practice.
- Angle preservation: they are conformal wherever the derivative is non-zero, so they preserve the magnitude of angles between smooth curves.
- Circle and line preservation: they map generalized circles, meaning circles and straight lines, to generalized circles. A line may become a circle and a circle may become a line.
- Action on infinity: points where c·z + d = 0 are sent to infinity, and the point at infinity itself may map to a finite point when c ≠ 0.
- Hyperbolic geometry connections: when restricted to domains such as the unit disk or upper half-plane, suitable Möbius transformations act as symmetries of important hyperbolic metrics.
These properties are easiest to understand by experimenting. If you keep c = 0, the map behaves much like the complex functions you already know from multiplication and addition. Once c becomes non-zero, the denominator starts to matter, and the image can stretch or swing dramatically as z approaches the pole. That contrast between smooth affine motion and inversion-like distortion is a big part of what this calculator helps you see.
How to Use the Möbius Transformation Calculator
This calculator applies the formula w = (a·z + b)/(c·z + d) to a single complex input point. Enter the coefficients and the point z, then press the transform button. The result area returns either a finite complex value or the word Infinity if the denominator is exactly zero.
Entering complex numbers
- Enter values in a+bi style, such as
1+2i,-0.5+3i,2-i,3,i, or-i. - The imaginary unit should be written as i.
- This page also supports simple parenthesized arithmetic such as
(1+i)/2, which is useful for neat exact-looking inputs.
What each input means
The four coefficients define the transformation itself. Think of a and b as controlling the numerator, while c and d control the denominator and therefore the location of any pole. The input z is the point you want to map. Because the formula is a ratio, changing c or d often has a bigger qualitative effect than beginners expect: it can move the singularity and completely reshape how nearby points behave.
Required fields and validity
- All four coefficients a, b, c, and d, plus the input point z, should be provided for a meaningful result.
- The conceptual theory assumes a·d − b·c ≠ 0. If this determinant is zero, the map is degenerate and no longer behaves like a genuine invertible Möbius transformation.
- If the calculator detects that c·z + d = 0, the image is the point at infinity and is reported that way rather than as an ordinary complex number.
Interpreting the Result
After you enter a, b, c, d, and z, the tool computes the image point w. The most direct interpretation is algebraic: it is the simplified value of the ratio after complex multiplication and addition are carried out.
There is also a geometric interpretation. The output tells you where the point z lands after the transformation acts on the plane. If you try several nearby values of z, you can begin to see whether the map behaves mostly like a translation and rotation, mostly like an inversion, or like a combination of both. If the output suddenly becomes extremely large or jumps to infinity, that is a sign that the denominator c·z + d is close to zero. In other words, you are probing the pole of the transformation.
This is one reason Möbius transformations are so useful in learning settings. A single numerical result can hint at a larger geometric story. The more sample points you test, the easier it becomes to predict which circles, lines, or regions may bend, flip, or stretch under the same coefficients.
Worked Example
Example 1: Simple rotation and scaling
Take
- a = 1 + i,
- b = 0,
- c = 0,
- d = 1,
- z = 1.
Here the formula simplifies to
w = (a·z + b)/(c·z + d) = ((1 + i)·1 + 0)/1 = 1 + i.
Algebraically, you are just multiplying by 1 + i. Geometrically, multiplication by 1 + i scales lengths by √2 and rotates the plane by 45°. So the point z = 1 moves to a point with equal real and imaginary parts and larger magnitude.
Example 2: Why the determinant condition matters
Now consider
- a = 1,
- b = 0,
- c = 1,
- d = 0,
- z = 2.
Then
w = (1·z + 0)/(1·z + 0) = z/z = 1 for any non-zero z.
This is intentionally a degenerate example. The determinant is ad − bc = 1·0 − 0·1 = 0, so the coefficients do not define a genuine non-degenerate Möbius transformation. Instead of producing an invertible map, they collapse every non-zero input to the same output. It is a helpful cautionary example because it shows exactly why the determinant condition is not optional.
Example 3: A genuine inversion
For a more typical inversion-like map, take
- a = 0,
- b = 1,
- c = 1,
- d = 0,
- z = 2 + i.
Now
w = (0·z + 1)/(1·z + 0) = 1/z.
To compute this explicitly, write
1/(2 + i) = (2 − i) / ((2 + i)(2 − i)) = (2 − i)/(4 + 1) = (2 − i)/5.
So the image of z = 2 + i under w = 1/z is w = (2 − i)/5. Geometrically, this behaves like inversion in the unit circle together with reflection across the real axis. It is a classic example because it makes the denominator’s role impossible to miss.
Comparison of Common Special Cases
The table below summarizes several important special forms of Möbius transformations and how they act on points in the complex plane.
| Type | Coefficients (a, b, c, d) | Formula for f(z) | Geometric effect |
|---|---|---|---|
| Identity map | a = 1, b = 0, c = 0, d = 1 | f(z) = z | Leaves every point fixed and serves as a reference case. |
| Pure translation | a = 1, b = b₀, c = 0, d = 1 | f(z) = z + b₀ | Shifts the whole plane by a fixed complex vector. |
| Rotation and dilation | a = λ, b = 0, c = 0, d = 1 | f(z) = λ·z | Scales by |λ| and rotates by arg(λ). |
| Inversion in the unit circle | a = 0, b = 1, c = 1, d = 0 | f(z) = 1/z | Sends circles and lines to circles or lines and swaps inside and outside behavior away from the boundary. |
| General case | arbitrary a, b, c, d with a·d − b·c ≠ 0 | f(z) = (a·z + b)/(c·z + d) | Combines translation, rotation, scaling, and inversion while preserving generalized circles. |
Limitations and Assumptions
This calculator focuses on the transformed value of a single input point z. That makes it excellent for checking homework, testing guesses, and building intuition point by point, but it also means there are a few assumptions to keep in mind.
- Determinant condition: the standard theory assumes a·d − b·c ≠ 0. If this quantity is zero, the formula may collapse regions to points and should not be interpreted as an invertible Möbius map.
- Point at infinity: if c·z + d is exactly zero, the image is the point at infinity. The calculator represents that as text.
- Numeric precision: values extremely close to a pole can be sensitive to rounding. A tiny change in input may create a large visible change in output.
- No automatic plotting: the page computes images of individual points rather than drawing the full image of a circle, line, or region.
- Input format: results are only meaningful when the entered text is a valid complex value or simple complex expression.
Within those limits, the tool is well-suited for experimentation. In fact, trying a family of nearby points is often more revealing than a single formal proof when you are first learning how Möbius transformations behave near fixed points, poles, and inversion regions.
Matrix Representation and Group Structure
Every Möbius transformation can be represented, up to a non-zero scalar multiple, by a 2×2 complex matrix
M = [[a, b], [c, d]].
Composition of Möbius transformations corresponds to matrix multiplication. That is one of the cleanest reasons the subject feels unified: performing one transformation after another is the same as multiplying the associated matrices and reading off the new coefficients. Two matrices that differ by an overall non-zero complex factor represent the same Möbius map, so the true symmetry object is projective rather than ordinary linear algebra.
This viewpoint connects Möbius transformations to the group PSL(2, ℂ), the projective special linear group of 2×2 complex matrices with non-zero determinant modulo scalar multiples. For practical calculator use, you do not need the full group theory. Still, it explains why these transformations show up in geometry, dynamics, and mathematical physics: they are not just isolated formulas but part of a deep algebraic structure.
Mini-Game: Target the Image Point
This optional arcade-style mini-game turns the calculator’s inverse problem into a quick visual challenge. You control the input point z on the left complex plane, while the mapped point w = (az+b)/(cz+d) appears on the right. Your job is to steer w into glowing targets before the timer runs out. The twist is that the same algebra from the calculator is doing the work in real time, so the denominator, pole, and distortion are not just abstract warnings anymore.
Each short run uses a sequence of fresh non-degenerate transformations. Affine rounds feel smooth and familiar. Rounds with non-zero c can bend motion sharply, especially near the pole where cz + d gets small. If you want a fast way to build intuition for how coefficients reshape the plane, this is a surprisingly effective way to do it.
Educational takeaway: when the pole is nearby, tiny movements in z can create huge jumps in w. That is exactly what the denominator cz + d predicts.