3. Postdoctoral Program - MSRI

2.2 Circle
Theorem 2. Let r ∈ Q with r �= 0. Given a circle C: �z − z0� = r in the complex
plane and a vertical line L: Re z = 1 . Let f be the following Möbius transformation
2r
f: P 1 (C) → P 1 (C) defined by f(z) = (z0 − r)z + 1
,
z
that maps the line L into the circle C. Therefore for any circle C there exists a
rational distance set.
Proof. Consider the transformation f(z) = (z0−r)z+1
. z
Since f is a Möbius transformation, f preserves the map from lines to circles.
Therefore we can choose 3 points on the line, determine where f maps them, and
then determine the unique circle that passes through those three points. Picking
1 1 1 1 1
1
, − i , + i , we get f( 2r 2r 2r 2r 2r 2r ) = z0 + r, f( 1 1 − i 2r 2r ) = z0 − ir, f( 1 1 + i 2r 2r ) = z0 + ir.
These are three points on the circle �z − z0� = r.
Figure 1: Example of a line that maps to a circle.
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