---
title: "Conformal Maps and Mobius Transformations"
description: "See that an analytic map with nonzero derivative preserves angles, and use Mobius maps to move standard regions onto one another."
canonical: https://lightmysky.com/learn/mathematics/conformal-maps-and-mobius-transformations-mt_WtHCbAT4GI
source: https://lightmysky.com/learn/mathematics/conformal-maps-and-mobius-transformations-mt_WtHCbAT4GI.md
retrieved: 2026-09-12
---

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# Conformal Maps and Mobius Transformations

See that an analytic map with nonzero derivative preserves angles, and use Mobius maps to move standard regions onto one another.

Subject: Mathematics · Area: Complex Analysis · Ages 21 to 22
Page: https://lightmysky.com/learn/mathematics/conformal-maps-and-mobius-transformations-mt_WtHCbAT4GI

## Ready when they can

- Explain why a nonzero derivative makes a map angle-preserving
- Build a Mobius transformation carrying three named points to three others
- Map a half-plane onto a disc and use it to transfer a boundary-value problem

## Lesson: Keep angles and bend regions

Conformal maps keep the angles between curves. Wherever the derivative is nonzero, the map acts locally like a rotation plus a scaling, and both keep angles intact. Zoom in close and every conformal map looks like multiplication by a nonzero number. Where the derivative vanishes, angles can multiply: squaring doubles angles at the origin.

Mobius maps are ratios of linear functions and the friendliest conformal maps. They send circles and lines to circles and lines. Three points determine one completely, so you can engineer maps carrying chosen triples to chosen triples. Each one chains translations, scalings, rotations, and a single flip, all conformal where defined.

The classic move carries the upper half plane onto the unit disc through z minus i over z plus i. The real line, the old boundary, lands on the unit circle, and interior points land inside. Boundary problems travel along: solve on the disc, then pull back. Laplace problems ride especially well since harmonicity survives analytic composition.

**Tip.** Check the derivative before claiming angles. The exponential qualifies everywhere since its derivative never vanishes. Squaring fails at the origin alone. Map choice follows the boundary: lines beg for the half plane to disc move.

**Recap.** Nonzero slope keeps angles, three points fix Mobius, and hard shapes map to easy ones.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Matrix Multiplication and What It Represents](https://lightmysky.com/learn/mathematics/matrix-multiplication-and-what-it-represents-mt_0fJOMPlCxr)
- [The Argument Principle and Rouche's Theorem](https://lightmysky.com/learn/mathematics/the-argument-principle-and-rouches-theorem-mt_J1dC428M84)
- [Complex Differentiability and the Cauchy-Riemann Equations](https://lightmysky.com/learn/mathematics/complex-differentiability-and-the-cauchy-riemann-equations-mt_nZNmZcl9Ss)