- Hilbert's arithmetic of ends
Hilbert's arithmetic of ends is an algebraic approach introduced by German mathematician
David Hilbert forPoincaré disk model ofhyperbolic geometry [Hilbert, "A New Development of Bolyai-Lobahevskian Geometry" as Appendix III in "Foundations of Geometry", 1971.] . Hilbert defines a field of ends with amultiplicative distance function over the field. So, one can also set up a hyperbolicanalytic geometry andhyperbolic trigonometry , whereby any geometric problem can be translated into an algebraic problem in the field.In Poincaré model of hyperbolic plane, every
limiting parallel ray intersects at apoint lying on thelimit circle which is not included in thehyperbolic plane . So one may take aline as having uniquely two ends. Thus, anaddition and amultiplication over the set of these ends constructs a field [Robin Hartshorne, "Geometry: Euclid and Beyond", Springer-Verlag, 2000, sec. 41.] .A line of the geometry is represented by an ordered pair ("a", "b") of distinct elements of the field of ends. On contrary to cartesian plane of Euclidean geometry, a point is represented by a "point equation" which is satisfied by every line passing through the point, i.e. a
pencil of intersecting lines represents a unique point.Now, we will define the operations over the set of ends, "H".
Addition over ends
Definition.:"Given two ends α, β not equal to scriptstyleinfty and any point C on the line scriptstyle(0,, infty). Let A be its
reflection in the line scriptstyle(alpha,, infty). Let B its reflection in the line scriptstyle(eta,, infty). Then α + β is the end of theperpendicular bisector of AB other than scriptstyleinfty."The addition is
well-defined , and makes the set ("H", +) anabelian group with additiveidentity 0.The addition is usually understood as
:sigma_{alpha+eta}=sigma_eta sigma_0 sigma_alpha,
where for any end α, σα denotes reflection in the line scriptstyle(alpha,,infty). Thus the addition is independent of the choice of "C".
The existence of such an addition operation is based on a theorem of three reflection.
Theorem.:"Given three lines a ,b, c in the hyperbolic plane with a common end ω, there exist a fourth line d with end ω such that reflection in d is equal to the product of the reflections in a, b, c:":::sigma_c sigma_b sigma_a = sigma_d:"where scriptstylesigma_ell, denotes the reflection in the line scriptstyleell,."
So it is clear that the addition is just taking "a", "b", "c" as α, 0, β and "d" as α + β respectively.
Note that an "end" is an
equivalence class of limiting parallel rays. One can fix a hyperbolic line and label its ends 0 and scriptstyleinfty. Here "H" is the set of all ends in the plane different from scriptstyleinfty, then one sets scriptstyle H',=,H, cup, {infty}, so that scriptstyle H' is the set of all ends of the plane. We will make this set into an abelian field by defining also a multiplication on it.Multiplication over ends
The multiplication over the field is defined by fixing a line perpendicular to the line scriptstyle (0,infty) where they meet at the point "O", and label one of its ends 1, the other −1.
Definition.:"Given ends scriptstylealpha,, eta, the lines scriptstyle(alpha,,-alpha) and scriptstyle (eta,,-eta) meet the line scriptstyle(0,,infty) with right angles at A and B, respectively.:So "C" at where the line scriptstyle(alpha eta,,-alpha eta) is perpendicular to scriptstyle(0,,infty), is the point which satisfies the relation,
:: BA' = OC ,
:where the point "A' "is the reflection of "A" respect to the point "O.
In other words, the point "C" satisfies the relation "OA + OB = OC", according to the euclidean segment addition. So the field has an additive multiplication over line segments. It makes scriptstyle(H, setminus, {0},ullet) an abelian group with identity 1.
Rigid motions
Let scriptstyle Pi be a
hyperbolic plane and "H" its field of ends, as introduced above. In the plane scriptstyle Pi, we haverigid motion s and their effects on ends as follows:* The reflection in scriptstyle(0,, infty) sends scriptstyle x, in, H' to −"x".
::x'=-x.,
* The reflection in (1, −1) gives,
::x'={1 over x}.,
*
Translation along scriptstyle(0,,infty) that sends "1" to any scriptstyle a, in, H, "a" > 0 is represented by::x'=ax.,
* For any scriptstyle a, in, H, there is a rigid motion σ(1/2)"a" σ0, the composition of reflection in the line scriptstyle(0,infty) and reflection in the line scriptstyle((1/2) a,, infty), which is called rotation around scriptstyle infty is given by
::x'=x+a.,
* The
rotation around the point "O", which sends 0 to any given end scriptstyle a, in, H, effects as::x'=frac{x+a}{1-ax}
:on ends. The rotation around "O" sending 0 to scriptstyle infty gives
::x'=-{1 over x}.
References
Wikimedia Foundation. 2010.