Möbius Transformation Calculator
What Is a Möbius Transformation?
A Möbius transformation (also called a linear fractional transformation) is a complex function of the form
w = f(z) = ,
where a, b, c, d, and the variable z are complex numbers. Möbius transformations act on the extended complex plane (the complex numbers plus the point at infinity) and are central objects in complex analysis, conformal mapping, and hyperbolic geometry.
Geometrically, these maps are angle-preserving (conformal) and send lines and circles in the complex plane to other lines or circles. They are flexible enough to describe many useful operations, yet structured enough to have strong algebraic properties.
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.
A fundamental non-degeneracy condition is that
a·d − b·c ≠ 0.
When this determinant is non-zero, the transformation is invertible and its inverse is again a Möbius transformation.
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, z4, the cross-ratio is
((z₁ − z₃)(z₂ − z₄)) / ((z₁ − z₄)(z₂ − z₃)).
Preservation of the cross-ratio characterizes Möbius transformations and links them to projective geometry.
Matrix Representation and Group Structure
Every Möbius transformation can be represented (up to a non-zero scalar factor) by a 2×2 complex matrix
M = [[a, b], [c, d]].
Composition of Möbius transformations corresponds to matrix multiplication. Two matrices that differ by an overall non-zero complex factor represent the same Möbius map, so the set of all Möbius transformations can be identified with the group PSL(2, ℂ), the projective special linear group of 2×2 complex matrices with non-zero determinant modulo scalar multiples.
Geometric Properties and Interpretation
Möbius transformations have several key geometric properties:
- Angle preservation: they are conformal wherever the derivative is non-zero, so they preserve the magnitude of angles between smooth curves.
- Circle/line preservation: they map generalized circles (circles and straight lines) to generalized circles. A line can become a circle and vice versa.
- Action on infinity: points where c·z + d = 0 are sent to infinity, and the point at infinity can be mapped back to a finite point depending on the coefficients.
- Hyperbolic isometries: when restricted to certain domains, such as the unit disk or upper half-plane, suitable Möbius transformations act as isometries of the corresponding hyperbolic metric.
By adjusting a, b, c, and d, you can combine familiar operations such as translations, scalings, rotations, and inversions into a single unified framework.
How to Use the Möbius Transformation Calculator
This calculator applies the formula w = (a·z + b)/(c·z + d) to the complex number z using your chosen coefficients a, b, c, and d.
Entering complex numbers
- Enter all complex numbers in a+bi format, for example:
1+2i,-0.5+3i,2-i, or3(for a purely real value). - The imaginary unit should be written as i.
- You may use basic arithmetic like
(1+i)/2if supported by the underlying math library.
Required fields and validity
- All four coefficients a, b, c, d and the input point z should be provided for a meaningful result.
- The transformation is considered non-degenerate only when a·d − b·c ≠ 0.
- If the calculator detects that c·z + d = 0, the corresponding image is the point at infinity; it may be reported textually instead of as a finite complex number.
Interpreting the Result
After you enter a, b, c, d, and z, the tool computes
w = (a·z + b)/(c·z + d).
You can interpret the output in several ways:
- Algebraic: treat w as the complex number obtained by substituting your chosen values into the formula and simplifying.
- Geometric: view w as the image of the point z under a combination of translation, scaling, rotation, and inversion encoded by the coefficients.
- Structural: by testing several input values of z, you can build intuition about how an entire region (such as a circle, line, or disk) is mapped under the transformation.
If the calculator displays a message that the image is at infinity or numerically very large, it means that the denominator c·z + d is zero or extremely close to zero for the chosen values.
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 multiply 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 the point at a 45° angle from the positive real axis with magnitude √2.
Example 2: Inversion-type behavior
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. In this special case, the transformation sends every non-zero point to 1 and sends z = 0 to 0/0, which is undefined. This illustrates how certain coefficient choices can collapse regions and why the determinant condition a·d − b·c ≠ 0 is important for invertibility.
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 is an inversion in the unit circle combined with reflection across the real axis.
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; used 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 b₀. |
| 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; exchanges inside and outside of the unit circle (excluding the boundary). |
| General case | arbitrary a, b, c, d with a·d − b·c ≠ 0 | f(z) = (a·z + b)/(c·z + d) | Combination of translation, rotation, scaling, and inversion; maps generalized circles to generalized circles. |
Limitations and Assumptions
This calculator focuses on computing the complex value w for a single input z, given coefficients a, b, c, and d. Keep in mind the following assumptions and limitations when interpreting the results:
- Determinant condition: The conceptual theory usually assumes a·d − b·c ≠ 0. If this quantity is zero, the transformation becomes degenerate and may collapse regions to points or lines.
- Point at infinity: If the denominator c·z + d is exactly zero, the image is the point at infinity. The tool may represent this as a special text value rather than a finite complex number.
- Numeric precision: All computations rely on the precision and parsing rules of the underlying math library. For coefficients or inputs with very large magnitude or extremely close to singular values, rounding errors can affect the displayed result.
- No geometric plotting: The calculator reports only the transformed value of individual points; it does not draw images of circles, lines, or regions under the transformation.
- Input format: Results are only meaningful if inputs follow the expected complex-number syntax (such as
a+bi). Unusual formatting may cause parse errors or unexpected behavior.
Within these limits, the tool is well-suited for experimenting with Möbius transformations numerically, checking hand calculations, and building intuition about how different coefficient choices act on sample points in the complex plane.