In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle. The group of orientation preserving isometries of the disk model is given by the projective special unitary group PSU(1,1), the quotient of the special unitary group SU(1,1) by its center {I, −I}. 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.
History The disk model was first described by Bernhard Riemann in an 1854 lecture (published 1868), which inspired an 1868 paper by Eugenio Beltrami. Henri Poincaré employed it in his 1882 treatment of hyperbolic, parabolic and elliptic functions, but it became widely known following Poincaré's presentation in his 1905 philosophical treatise, Science and Hypothesis. There he describes a world, now known as the Poincaré disk, in which space was Euclidean, but which appeared to its inhabitants to satisfy the axioms of hyperbolic geometry:"Suppose, for example, a world enclosed in a large sphere and subject to the following laws: The temperature is not uniform; it is greatest at their centre, and gradually decreases as we move towards the circumference of the sphere, where it is absolute zero. The law of this temperature is as follows: If R {\displaystyle R} be the radius of the sphere, and r {\displaystyle r} the distance of the point considered from the centre, the absolute temperature will be proportional to R 2 − r 2 {\displaystyle R^{2}-r^{2}} . Further, I shall suppose that in this world all bodies have the same co-efficient of dilatation, so that the linear dilatation of any body is proportional to its absolute temperature. Finally, I shall assume that a body transported from one point to another of different temperature is instantaneously in thermal equilibrium with its new environment. ... If they construct a geometry, it will not be like ours, which is the study of the movements of our invariable solids; it will be the study of the changes of position which they will have thus distinguished, and will be 'non-Euclidean displacements,' and this will be non-Euclidean geometry. So that beings like ourselves, educated in such a world, will not have the same geometry as ours." (pp.65-68)Poincaré's disk was an important piece of evidence for the hypothesis that the choice of spatial geometry is conventional rather than factual, especially in the influential philosophical discussions of Rudolf Carnap and of Hans Reichenbach.
Lines and distance
Hyperbolic straight lines or geodesics 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. 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, that is, so that |aq| > |ap| and |pb| > |qb|. The hyperbolic distance between p and q is then
d ( p , q ) = ln | a q | | p b | | a p | | q b | . {\displaystyle d(p,q)=\ln {\frac {\left|aq\right|\,\left|pb\right|}{\left|ap\right|\,\left|qb\right|}}.}
The vertical bars indicate Euclidean length of the line segment connecting the points between them in the model (not along the circle arc); ln is the natural logarithm. Equivalently, if u and v are two vectors in real n-dimensional vector space Rn with the usual Euclidean norm, both of which have norm less than 1, then we may define an isometric invariant by
δ ( u , v ) = 2 ‖ u − v ‖ 2 ( 1 − ‖ u ‖ 2 ) ( 1 − ‖ v ‖ 2 ) , {\displaystyle \delta (u,v)=2{\frac {\lVert u-v\rVert ^{2}}{(1-\lVert u\rVert ^{2})(1-\lVert v\rVert ^{2})}}\,,}
where ‖ ⋅ ‖ {\displaystyle \lVert \cdot \rVert } denotes the usual Euclidean norm. Then the distance function is
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