Möbius Transformation Calculator
Introduction to Möbius Transformations
A Möbius transformation calculator is most useful when you want to see how the ratio w = (az+b)/(cz+d) behaves for an actual complex input instead of only on paper. Enter the four coefficients a, b, c, and d together with a point z, and the page computes the image w. If the denominator is zero, the result is reported as infinity so the pole is easy to spot.
For complex analysis homework, the calculator works as a fast algebra check and a geometry refresher at the same time. The values you enter are complex coordinates rather than physical measurements, so the important question is whether the numerator and denominator are assembled consistently. If your coefficients come from an applied model, make sure every term belongs on the same complex scale before you trust the output.
The sections below unpack the formula, explain what each coefficient does, show why ad - bc must stay non-zero, and interpret poles, inversions, and common special cases. A worked example follows, and the optional mini-game at the end lets you steer points through live Möbius mappings and watch the images move in real time.
What a Möbius Transformation Does
A Möbius transformation is a linear fractional map of the extended complex plane, and this calculator evaluates one such map from the coefficients you provide.
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 pictured as the Riemann sphere, which is why Möbius maps are so central in complex analysis: they describe motions on a sphere-like compactification of the plane rather than on a flat grid alone.
One reason these maps appear so often is that they can look simple while still combining several effects. If c = 0, the formula reduces to the affine map w = az + b, so the behavior is just multiplication followed by translation. When c ≠ 0, the denominator introduces inversion-like behavior and creates a pole at the point where cz + d = 0. Near that pole, the image can change very quickly, which is exactly the behavior this calculator helps you inspect.
Geometrically, these maps are conformal wherever the derivative is non-zero, and they send generalized circles to generalized circles. In this setting, a generalized circle means either a usual circle or a straight line. So even though the algebra is a ratio, the geometry is highly organized rather than arbitrary.
Möbius Transformation Formula
The formula used by this calculator is the same one above, repeated here because it is the key step in the computation.
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 case is usually shown as a label instead of 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 usually treat ad − bc ≠ 0 as part of the definition rather than as an extra warning.
MathML version of the Möbius formula
The calculator keeps the same expression available in MathML for readers and software that prefer structured math markup:
Cross-ratio view of the same Möbius map
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 viewpoint matters because it explains why Möbius transformations sit at the meeting point of complex analysis and projective geometry. If you know 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 map, you can usually determine the transformation uniquely.
Geometric Properties and Interpretation
Möbius transformations have a few geometric habits that make the calculator useful 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 feel when you experiment with different coefficients. If you keep c = 0, the map behaves like multiplication plus translation. Once c becomes non-zero, the denominator matters, and the image can swing or stretch dramatically as z approaches the pole. That contrast between affine motion and inversion-like distortion is one of the main things this calculator reveals.
How to Use the Möbius Transformation Calculator
This Möbius transformation calculator applies w = (a·z + b)/(c·z + d) to one complex point at a time. Enter the coefficients and the input 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 in the Möbius formula
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 larger geometric 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 Möbius 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 calculator returns the image point w. Algebraically, that means it simplifies the complex ratio after the multiplication and addition are carried out.
Geometrically, the output tells you where z lands under the Möbius map. Try several nearby values of z and you can start to see whether the coefficients behave mostly like a translation and rotation, mostly like an inversion, or like a blend of both. If the output suddenly becomes very large or jumps to infinity, that is a sign that c·z + d is close to zero, so 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: tracing a complex point through the Möbius map
Example 1: Rotation and scaling when c = 0
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: A degenerate coefficient set to avoid
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 true inversion with z = 2 + i
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
This table collects a few Möbius patterns that show up again and again when you test the calculator. Each row keeps the algebra simple enough to recognize the geometry at a glance.
| 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 for Möbius calculations
This Möbius transformation calculator evaluates one input point at a time, so its limits are mostly about algebraic validity, numerical stability, and the fact that it does not draw a full domain image.
- Determinant condition: the standard theory assumes a·d − b·c ≠ 0. If this quantity is zero, the coefficients 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
Behind every Möbius output on this page is a 2×2 complex matrix representation of the map you entered
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 Möbius transformation calculator into a live steering exercise. 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 same algebra driving the calculator is working in real time, so the denominator, pole, and distortion are no longer abstract warnings.
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.