Möbius Transformations: Circles to Circles Explained

A Möbius transformation is a map of the form w = (az + b)/(cz + d), where a, b, c, d are complex numbers with ad − bc ≠ 0. Its signature miracle: it sends circles and straight lines to circles and straight lines (a line counts as a circle passing through infinity).

These maps matter because they are the most symmetric transformations of the complex plane — they form a group under composition, preserve the cross-ratio of four points, and give all conformal automorphisms of the disc and half-plane. They appear in hyperbolic geometry, relativity, and even electrical engineering's Smith chart.

Key Concepts

• General form: w = (az + b)/(cz + d) with ad − bc ≠ 0; scaling all four coefficients by the same non-zero number gives the same map, so there are 3 complex (6 real) degrees of freedom.
• Extended plane: the map is a bijection of the Riemann sphere: z = −d/c maps to ∞ and z = ∞ maps to a/c.
• Circle-to-circle: every circle or line maps to a circle or line.
• Three-point determination: a Möbius map is uniquely fixed by the images of any three distinct points.
• Fixed points: solving w = z gives a quadratic, so a non-identity Möbius map has one or two fixed points.
• Cross-ratio: (z, z₁, z₂, z₃) = (w, w₁, w₂, w₃) is preserved by every Möbius transformation.

Formulas

• Definition: w = (az + b)/(cz + d), ad − bc ≠ 0.
• Inverse: z = (dw − b)/(−cw + a), also a Möbius transformation.
• Composition: composing two Möbius maps corresponds to multiplying their 2×2 coefficient matrices.
• Disc automorphism: w = e^{iθ}(z − a)/(1 − āz) with |a| < 1 maps the unit disc onto itself.
• Half-plane to disc: w = (z − i)/(z + i) maps the upper half-plane Im(z) > 0 onto the unit disc |w| < 1.
• Building blocks: every Möbius map is a composition of translations (z + b), dilations/rotations (az), and inversion (1/z).

Worked Example

Find the Möbius transformation sending 0 → 1, 1 → i, and ∞ → −1.

1. w(∞) = a/c = −1, so a = −c.
2. w(0) = b/d = 1, so b = d.
3. w(1) = (a + b)/(c + d) = i. Substituting a = −c, b = d: (−c + d)/(c + d) = i.
4. Solve: −c + d = ic + id, so d(1 − i) = c(1 + i), giving c/d = (1 − i)/(1 + i) = −i.
5. Choose d = 1: then b = 1, c = −i, a = i. Check ad − bc = i(1) − 1(−i) = 2i ≠ 0. ✓
6. Verify: w(z) = (iz + 1)/(−iz + 1); w(1) = (1 + i)/(1 − i) = i ✓, w(0) = 1 ✓, w(∞) = i/(−i) = −1 ✓.

Common Misconceptions

• "Möbius maps send circles to circles but lines to lines." Both go to the unified family: a line is a circle through ∞, and inversion can turn a line into a genuine circle.
• "Every Möbius map has two fixed points." Parabolic maps like w = z + 1 have exactly one fixed point (at ∞); the quadratic w = z degenerates in that case.
• "The map is undefined at its pole." On the Riemann sphere it is perfectly defined: z = −d/c maps to ∞, keeping the map bijective.

Real-World Applications

• Hyperbolic geometry: disc automorphisms are the orientation-preserving isometries of the Poincaré disc model.
• Special relativity: Lorentz transformations act as Möbius maps on the celestial sphere of light rays.
• Electrical engineering: the Smith chart is a Möbius map turning impedance calculations into circle geometry.
• Computer graphics: perspective projections are Möbius-like maps sending lines to lines.

Möbius transformations are the circle-preserving bijections of the extended complex plane, fixed by three point-images and decomposable into translations, scalings, and inversion. The worked example shows the standard solve — match a/c, b/d, and one finite image — and the same pattern builds disc automorphisms, half-plane maps, and the Smith chart.