Transferring the Poincar\'e unit disc to a unit sphere with elliptic geometry, suggests simply identifying antipodal Points, Circles etc: a circle outside the disc now represents the same Circle as its inverse with respect to Infinity inside the disc. Note that this strategem requires us to regard the pair of Curves equidistant from either side of a Line as a single Equidistant Curve. The hyperbolic version of Soddy now may be extended to include as special cases of Circles: Equidistant Curves, Infinity, Horocycles and Lines (but not Points). In particular, 3 tangent Lines have a coincident pair of tangent Circles; 3 tangent Horocycles have 1 tangent Circle, together with Infinity. Assuming unit space curvature, Curvature of a Circle exceeds unity; of a Horocycle equals unity; of an Equidistant Curve inceeds unity; of a Line equals zero; and Ultracycles are now identified with Circles. WFL On 2/14/08, Eugene Salamin <gene_salamin@yahoo.com> wrote:

----- Original Message ---- From: Fred lunnon <fred.lunnon@gmail.com>

To: math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, February 13, 2008 4:03:45 PM Subject: Re: [math-fun] Homotheties of the hyperbolic plane

... Which reminds me of another similarly off-the-wall query. In the Poincar\'e model ("circle" including line and point in Euclidean plane) circles within the disc correspond to Circles; circles orthogonal to the disc correspond to Lines; circles tangent to the disc correspond to Horocycles; circles meeting to the disc correspond to Equidistant Curves; but circles outside the disc correspond to what in Hyperbolic space?

WFL

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There is one more circle, the boundary of the disk. It is infinity, the set of directions in the hyperbolic plane. As for the "outside", it seems to be a parallel universe, another hyperbolic plane. On the Poincare sphere, pick a circle, call it "infinity". It separates the sphere into two caps, which can be made into two hyperbolic planes. Conformal maps on the sphere which preserve infinity become congruences in the hyperbolic planes. Reflection through infinity exchanges the two universes and establishes a correspondence between their objects. A circle orthogonal to infinity reflects onto itself, so its two hyperbolic lines correspond. On the other hand, a circle intersecting infinity nonorthogonally reflects onto a different circle, and so its two equidistant curves are noncorresponding. No doubt much more can be said. In particular, many parallel universes can coexist on the Poincare sphere, and be related via groups of conformal mappings.

Gene

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