Möbius transformations map circles and lines to circles and lines

Builds on Möbius transformation Riemann sphere

Statement

A generalized circle in the Riemann sphere $\widehat{\mathbb{C}}$ is either a circle in $\mathbb{C}$ , or a set $L \cup \{\infty\}$ where $L$ is a line in $\mathbb{C}$ . Let $f$ be a Möbius transformation. Then for each generalized circle $C$ , the image $f(C)$ is a generalized circle.

Remarks

A circle can go to a line: the map $z \mapsto \frac{1}{z}$ sends the circle $|z - 1| = 1$ , which contains $0$ , to the line $\operatorname{Re} w = \frac{1}{2}$ together with $\infty$ . The image of a generalized circle is a line together with $\infty$ exactly when the generalized circle contains the point that $f$ sends to $\infty$ . The proof checks the result for $z \mapsto z + b$ , $z \mapsto az$ and $z \mapsto \frac{1}{z}$ , because each Möbius transformation is a composition of such maps.

Reading path

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Complex numbers object
Complex plane concept
Riemann sphere definition
Möbius transformation definition
Möbius transformations map circles and lines to circles and lines statement

Tag	`0SXDHQM2MH`
URI	`m2://0SXDHQM2MH`
Aliases	`mobius-maps-preserve-generalized-circles`
Kind	statement, theorem
Writing	W0 stub
MSC	30C35
Status	This copy of the site has no build data. Run `m2 status 0SXDHQM2MH`.
Uses	Möbius transformation, Riemann sphere

Möbius transformations map circles and lines to circles and lines

Statement

Remarks

Reading path

Further reading

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