- 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 and any point C on the line . Let A be its
reflection in the line . Let B its reflection in the line . Then α + β is the end of theperpendicular bisector of AB other than ."The addition is
well-defined , and makes the set ("H", +) anabelian group with additiveidentity 0.The addition is usually understood as
:
where for any end α, σα denotes reflection in the line . 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:"::::"where denotes the reflection in the line ."
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 . Here "H" is the set of all ends in the plane different from , then one sets , so that 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 where they meet at the point "O", and label one of its ends 1, the other −1.
Definition.:"Given ends , the lines and meet the line with right angles at A and B, respectively.:So "C" at where the line is perpendicular to , is the point which satisfies the relation,
::
: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 an abelian group with identity 1.
Rigid motions
Let be a
hyperbolic plane and "H" its field of ends, as introduced above. In the plane , we haverigid motion s and their effects on ends as follows:* The reflection in sends to −"x".
::
* The reflection in (1, −1) gives,
::
*
Translation along that sends "1" to any , "a" > 0 is represented by::
* For any , there is a rigid motion σ(1/2)"a" σ0, the composition of reflection in the line and reflection in the line , which is called rotation around is given by
::
* The
rotation around the point "O", which sends 0 to any given end , effects as::
:on ends. The rotation around "O" sending 0 to gives
::
References
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