Poincaré disc

In geometry, the Poincaré disk model also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all segments of circles contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.

Along with the Klein model and the Poincaré half-space model, it was proposed by Eugenio Beltrami who used these models to show that hyperbolic geometry was equiconsistent with Euclidean geometry. It is named after Henri Poincaré, because his rediscovery of this representation fourteen years later became better known than the original work of Beltrami.

The Poincaré ball model is the similar model for 3 or n-dimensional hyperbolic geometry in which the points of the geometry are in the n-dimensional unit ball.

Hyperbolic Straight lines consist of all arcs of Euclidean circles contained within the disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.

The unique hyperbolic line through two points P and Q not on a diameter of the boundary circle can be constructed by:

If P and Q are on a diameter of the boundary circle that diameter is the hyperbolic line.

Another way is:

Distances in this model are Cayley–Klein metrics. Given two distinct points p and q inside the disk, the unique hyperbolic line connecting them intersects the boundary at two ideal points, a and b, label them so that the points are, in order, a, p, q, b and |aq| > |ap| and |pb| > |qb|.

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