Möbius transformations and ellipses
نویسندگان
چکیده
If T : C∪{∞} → C∪{∞} 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 (E) of a non-circular ellipse E ⊆ C, although as shown in Figure 1, such a curve is not always an ellipse. For the sake of convenience, we will call a curve C a “möte,” for “Möbius Transformation of an Ellipse,” if C = T (E) for some non-circular ellipse E and Möbius transformation T . We will also call two curves C1 and C2 in C ∪ {∞} “Möbius equivalent” if there exists a Möbius transformation T such that C2 = T (C1). Our main result is that two mötes, T1(E1) and T2(E2), are Möbius equivalent if and only if E1 and E2 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.
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