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Möbius transformations and ellipses. (English)
Pi Mu Epsilon J. 12, No. 6, 339-345 (2007).
Summary: If $T: \Bbb C\cup\{\infty\}\to\Bbb C\cup\{\infty\}$ is a Möbius transformation of the extended complex plane, it is well-known that the image under $T$ of a line or circle is another line or circle. It seems natural to consider the image $T({\cal E})$ of a non-circular ellipse ${\cal E}\subseteq \Bbb C$, although such a curve is not always an ellipse. We will call a curve $\cal C$ a “möte”, for “Möbius Transformation of an Ellipse”, if ${\cal C} = T({\cal E})$ for some non-circular ellipse $\cal E$ and Möbius transformation $T$. We will also call two curves ${\cal C}_1$ and ${\cal C}_2$ in $\Bbb C \cup \{\infty\}$ “Möbius equivalent” if there exists a Möbius transformation $T$ such that ${\cal C}_2 = T({\cal C}_1)$. Our main result is that two mötes, $T_1({\cal E}_1)$ and $T_2({\cal E}_2)$, are Möbius equivalent if and only if ${\cal E}_1$ and ${\cal E}_2$ are ellipses with the same eccentricity. In this sense, the eccentricity is an invariant of an ellipse not only under similarity transformations of the plane, but also under the larger group of Möbius transformations. In the last section we briefly consider some other special plane curves in the extended complex plane.
Classification: I80 G90