A Family of Compactifications Accounting for All Arguments of Infinity
نویسندگان
چکیده
We study a family of compactifications of the complex plane that distinguishes among the values of positive infinity, negative infinity, and other “arguments” of infinity. We augment the complex plane with an ideal set of points on an ideal circle. Each point on this circle corresponds to a different ray in the complex plane emanating from the origin. In this manner we obtain the “ultra extended complex plane.” The set of points in the ultra extended complex plane maps to a bowl-shaped subset of the Riemann sphere via a certain projection. We obtain the Riemann stereographic projection as a degenerate limit of a family of projections. Thus, we demonstrate how the infinitely many different directions at infinity degenerate into a single ideal point at infinity that is added to R2 in order to produce the extended complex plane. The features of this mapping are studied; the introduction of a metric on the ultra extended complex plane is a focal subject of this paper.
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