Complex Mappings and Transformations (Translations, Rotations, Scalings)
A complex function w = f(z) maps each point of the z-plane to a point of the w-plane, and the simplest mappings are the translation w = z + b, the rotation w = eiαz and the scaling w = kz, which together form the similarities w = az + b. In brief: w = (1 + i)z + 2 rotates by 45°, scales by √2 and then shifts by 2. The mapping w = 1/z is an inversion: it maps circles and lines to circles and lines. The mappings preserve angles (they are conformal) wherever the derivative is nonzero.
Real functions are drawn as graphs. Complex functions need four dimensions for a graph, so we study how they transform regions of the plane. This shows in a very visual way what the function does. This guide explains translation, rotation and scaling, the combination w = az + b, the inversion w = 1/z, the square w = z2 and the Möbius transformations, with worked examples of images of lines and circles and a table of mappings.
What are translations, rotations and scalings?
- Translation w = z + b: every point moves by the vector b.
- Rotation w = eiαz (a number of modulus 1): rotates by the angle α about the origin. For α = 90° this is w = iz.
- Scaling w = kz with real k > 0: a dilation about the origin with the factor k.
- Reflection w = z̄: the mirror image in the real axis (not a function of z in the analytic sense, but a useful mapping).
The multiplication by a complex number combines a rotation and a scaling.
Example 1
The mapping w = iz maps the point 2 + i to i(2 + i) = −1 + 2i, a rotation by 90°. The square with vertices 0, 1, 1 + i, i is mapped to the vertices 0, i, −1 + i, −1, which is the same square rotated by 90°.
What is the mapping w = az + b?
Writing a = ρeiα, the mapping w = az + b is a rotation by α, a scaling by ρ and a translation by b. It maps lines to lines, circles to circles and preserves angles and the ratios of lengths (a similarity transformation, orientation-preserving). It has a fixed point z0 = b/(1 − a) if a ≠ 1, and the mapping is a rotation by α and scaling by ρ about z0.
Example 2: the image of a circle
The circle |z| = 1 under w = 2z + 3 + i. The image has the center 3 + i and radius 2: |w − (3 + i)| = |2z| = 2.
Example 3: the image of a line
The line Re z = 1 under w = (1 + i)z. If z = 1 + it, then w = (1 + i)(1 + it) = (1 − t) + (1 + t)i. So u = 1 − t and v = 1 + t, so u + v = 2: a line.
Example 4: finding the mapping
Find the mapping w = az + b that maps 0 to 1 and 1 to i. From 0 ↦ 1: b = 1. From 1 ↦ i: a + 1 = i, so a = −1 + i. So w = (−1 + i)z + 1. Check with z = 1: −1 + i + 1 = i ✓.
What does w = 1/z do?
The mapping w = 1/z = z̄/|z|2 sends a point with modulus r and argument θ to the point with modulus 1/r and argument −θ. The unit circle maps to itself (each point goes to its conjugate), the inside of the unit disc is mapped to the outside, and 0 corresponds to the “point at infinity”. Main properties:
- A circle not passing through the origin maps to a circle.
- A circle through the origin maps to a line (not through the origin).
- A line not through the origin maps to a circle through the origin.
- A line through the origin maps to a line through the origin (reflected in the real axis).
Example 5: the line Re z = 1
With z = 1 + it: w = 1/(1 + it) = (1 − it)/(1 + t2). So u = 1/(1 + t2), v = −t/(1 + t2). Then u2 + v2 = 1/(1 + t2) = u, so (u − 1/2)2 + v2 = 1/4: a circle with center 1/2 and radius 1/2 passing through the origin ✓.
Example 6: the circle |z − 1| = 1
This circle passes through 0, so its image is a line. The point z = 2 maps to 1/2 and z = 1 + i maps to (1 − i)/2. The image is the vertical line Re w = 1/2. Check: for z on the circle, |z|2 = 2 Re z (from |z − 1|2 = 1), so Re(1/z) = Re z/|z|2 = 1/2 ✓.
What does w = z2 do?
In polar form z = reiθ maps to r2e2iθ, so moduli are squared and angles are doubled. The upper half-plane (0 < θ < π) maps to the whole plane minus the positive real axis. The map is two-to-one: z and −z have the same image. The first quadrant maps to the upper half-plane. A line Re z = c maps to the parabola: z = c + it gives w = c2 − t2 + 2cti, so u = c2 − t2 and v = 2ct, which gives u = c2 − v2/(4c2), a parabola opening to the left. A circle |z| = r maps to the circle |w| = r2, traversed twice. The inverse of z2 is the square root.
Example 7
The image of the sector 0 ≤ arg z ≤ π/3, |z| ≤ 2 under w = z2 is the sector 0 ≤ arg w ≤ 2π/3, |w| ≤ 4. The point 1 + i (modulus √2, angle 45°) maps to 2i (modulus 2, angle 90°) ✓.
What are Möbius transformations?
A Möbius transformation has the form w = (az + b)/(cz + d) with ad − bc ≠ 0. It is a composition of translations, rotations, scalings and the inversion 1/z, so it maps circles and lines to circles and lines and preserves angles. Three points and their images determine the transformation uniquely. These include w = z + b, w = az and w = 1/z as special cases.
Example 8
w = (z − 1)/(z + 1) maps 1 to 0, −1 to ∞, 0 to −1 and i to (i − 1)/(i + 1) = i. The unit circle |z| = 1 maps to the imaginary axis, since the quotient is purely imaginary for |z| = 1, z ≠ −1. The right half-plane Re z > 0 maps to the unit disc |w| < 1: since |z − 1| < |z + 1| for Re z > 0, we get |w| < 1.
Example 9: a composition
w = 1/(z − 2) + 1 is the composition of a translation by −2, the inversion and a translation by 1. The line Re z = 2 passes through the point 2 where the inversion has its pole, so it maps to a line: z = 2 + it gives w = 1 + 1/(it) = 1 − i/t, the vertical line Re w = 1.
Table of mappings
| Mapping | Effect | Image of z = 1 + i | Image of the unit circle |
|---|---|---|---|
| w = z + 2 | translation | 3 + i | circle, center 2, radius 1 |
| w = iz | rotation 90° | −1 + i | unit circle |
| w = 2z | scaling × 2 | 2 + 2i | circle, radius 2 |
| w = (1 + i)z | rotation 45° and scaling × √2 | 2i | circle, radius √2 |
| w = 1/z | inversion | (1 − i)/2 | unit circle |
| w = z2 | squares the modulus, doubles the angle | 2i | unit circle (twice) |
| w = z̄ | reflection in the real axis | 1 − i | unit circle |
Where do these ideas come from?
August Ferdinand Möbius studied the transformations named after him in the 1850s, though they had been used earlier by others. Riemann’s work in the 1850s on conformal mappings, which preserve angles, showed that the open disc is equivalent to the upper half-plane, a result that became fundamental in complex analysis. Joukowski used a related mapping w = z + 1/z in 1910 to model the flow around airfoils.
Where are they used today?
Conformal mappings simplify problems in electrostatics, fluid flow and heat conduction by transforming complicated regions into simple ones. The Smith chart in radio engineering is a Möbius mapping of impedances to a unit disc. In computer graphics, Möbius transformations underlie projective transformations of the sphere, and in physics they describe Lorentz transformations of the celestial sphere.
Which mistakes are common?
Expecting a graph
A complex function is visualised by the images of regions, not by a graph in the plane.
Forgetting that z2 is two-to-one
z and −z have the same image, so there is no unique inverse without a choice.
Mixing up the mapping and its inverse
To find the image of a curve, substitute the parametrization of the curve into f. To find the preimage, solve f(z) = w for z.
Not checking the condition ad − bc ≠ 0
If ad = bc, the Möbius transformation is constant. For example, (2z + 2)/(z + 1) = 2 for all z ≠ −1.