Hilbert's arithmetic of ends

In mathematics, specifically in the area of hyperbolic geometry, Hilbert's arithmetic of ends is a method for endowing a geometric set, the set of ideal points or "ends" of a hyperbolic plane, with an algebraic structure as a field.

[1] In a hyperbolic plane, one can define an ideal point or end to be an equivalence class of limiting parallel rays.

In the Poincaré disk model or Klein model of hyperbolic geometry, every ray intersects the boundary circle (also called the circle at infinity or line at infinity) in a unique point, and the ends may be identified with these points.

For the purpose of Hilbert's arithmetic, it is expedient to denote a line by the ordered pair (a, b) of its ends.

The multiplication operation in the arithmetic of ends is defined (for nonzero elements x and y of H) by considering the lines (1,−1), (x,−x), and (y,−y).

, we have rigid motions and their effects on ends as follows: For a more extensive treatment than this article can give, confer.