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Abstract:

In inversive geometry, which deals with the space C ∪ {∞} and the group of Möbius transformations, the properties of circles are well-known. In particular, the image of a line or circle under a Möbius transformation is another line or circle. Some interesting questions arise when considering the actions of Möbius transformations on ellipses. Of course the circle could be considered a special case, but do any of its inversive properties generalize to non-circular ellipses? More specifically, the following Questions refer to E, an ellipse which is not a circle, contained in the subset C of the extended complex plane, C ∪ {∞}, and E ′ ⊆ C, another ellipse. Questions: I. What is the image of E under a Möbius transformation? II. Which Möbius transformations are symmetries of E? III. Which Möbius transformations T, ifany,havethe property T(E) ⊆E ′? From this point, we use the term “ellipse ” to mean only “non-circular ellipse, ” and the term “circle ” refers to both circles and extended lines (which include the point ∞). The Questions could be generalized to real conics in 1 general, and we briefly survey a few inversive properties of parabolas and hyperbolas. However, the geometry of the ellipse turns out to be a little more subtle, and it seems that the hyperbolas have already had their share of attention in the literature, which is one reason for an expository article focusing (!) on the ellipse. So, while the reader ponders these Questions about ellipses, we summarize some facts about Möbius transformations. Möbius transformations are maps T: C ∪ {∞} → C ∪ {∞}, of the form or T(Z)= aZ + b

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