intersects the boundary at two ideal points, a and b, label them so that the points are, in order, a, p, q, b and and . The hyperbolic distance between p and q is then: The vertical bars indicate Euclidean distances between the points in the model, log is the natural logarithm and the factor of one half is needed to give the model the standard curvature of −1. When one of the points is the origin and Euclidean distance between the points is r then the hyperbolic distance is: where artanh is the inverse hyperbolic function of the hyperbolic tangent. The Klein disk model In two dimensions the Beltrami–Klein model is called the Klein disk model. It is a disk and the inside of the disk is a model of the entire hyperbolic plane. Lines in this model are represented by chords of the boundary circle (also called the absolute). The points on the boundary circle are called ideal points; although well defined, they do not belong to the hyperbolic plane. Neither do points outside the disk, which are sometimes called ultra ideal points. The model is not conformal, meaning that angles are distorted, and circles on the hyperbolic plane are in general not circular in the model. Only circles that have their centre at the centre of the boundary circle are not distorted. All other circles are distorted, as are horocycles and hypercycles Properties Chords that meet on the boundary circle are limiting parallel lines. Two chords are perpendicular if, when extended outside the disk, each goes through the pole of the other. (The pole of a chord is an ultra ideal point: the point outside the disk where the tangents to the disk at the endpoints of the chord meet.) Chords that go through the centre of the disk have their pole at infinity, orthogonal to the direction of the chord (this implies that right angles on diameters are not distorted). Compass and straightedge constructions Here is how one can use compass and straightedge constructions in the model to achieve the effect of the basic constructions in the hyperbolic plane. The pole of a line. While the pole is not a point in the hyperbolic plane (it is an ultra ideal point) most constructions will use the pole of a line in one or more ways. For a line: construct the tangents to the boundary circle through